ńņš. 2 |

(Xj ā’ Xi )mi mj .

2

det Bm1 (X1 ) Bm2 (X2 ) . . . Bm (X ) = Xi j!

1ā¤i,j,ā¤n

i=1 j=1 1ā¤i<jā¤

(3.2)

As Alain Lascoux taught me, the natural environment for this type of determinants

is divided diļ¬erences and (generalized) discrete Wronskians. The divided diļ¬erence ā‚x,y

is a linear operator which maps polynomials in x and y to polynomials symmetric in x

and y, and is deļ¬ned by

f(x, y) ā’ f(y, x)

ā‚x,y f(x, y) = .

xā’y

Divided diļ¬erences have been introduced by Newton to solve the interpolation prob-

lem in one variable. (See [100] for an excellent introduction to interpolation, divided

diļ¬erences, and related matters, such as Schur functions and Schubert polynomials.) In

ADVANCED DETERMINANT CALCULUS 27

fact, given a polynomial g(x) in x, whose coeļ¬cients do not depend on a1, a2, . . . , am ,

Newtonā™s interpolation formula reads as follows (cf. e.g. [100, (Ni2)]),

g(x) = g(a1 ) + (x ā’ a1)ā‚a1 ,a2 g(a1) + (x ā’ a1)(x ā’ a2 )ā‚a2 ,a3 ā‚a1 ,a2 g(a1 )

+ (x ā’ a1 )(x ā’ a2 )(x ā’ a3 )ā‚a3,a4 ā‚a2 ,a3 ā‚a1 ,a2 g(a1) + Ā· Ā· Ā· . (3.3)

Now suppose that f1 (x), f2(x), . . . , fn (x) are polynomials in one variable x, whose

coeļ¬cients do not depend on a1, a2, . . . , an , and consider the determinant

det (fi (aj )). (3.4)

1ā¤i,j,ā¤n

Let us for the moment concentrate on the ļ¬rst m1 columns of this determinant. We

may apply (3.3), and write

fi (aj ) = fi (a1) + (aj ā’ a1)ā‚a1 ,a2 fi (a1) + (aj ā’ a1)(aj ā’ a2)ā‚a2 ,a3 ā‚a1 ,a2 fi (a1)

+ Ā· Ā· Ā· + (aj ā’ a1)(aj ā’ a2) Ā· Ā· Ā· (aj ā’ ajā’1 )ā‚ajā’1 ,aj Ā· Ā· Ā· ā‚a2 ,a3 ā‚a1,a2 fi (a1 ),

j = 1, 2, . . . , m1. Following [100, Proof of Lemma (Ni5)], we may perform column

reductions to the eļ¬ect that the determinant (3.4), with column j replaced by

(aj ā’ a1 )(aj ā’ a2 ) Ā· Ā· Ā· (aj ā’ ajā’1 )ā‚ajā’1 ,aj Ā· Ā· Ā· ā‚a2,a3 ā‚a1 ,a2 fi (a1),

j = 1, 2, . . . , m1, has the same value as the original determinant. Clearly, the product

k=1 (aj ā’ ak ) can be taken out of column j, j = 1, 2, . . . , m1 . Similar reductions can

jā’1

be applied to the next m2 columns, then to the next m3 columns, etc.

This proves the following fact about generalized discrete Wronskians:

Lemma 22. Let n be a nonnegative integer, and let Wm (x1 , x2, . . . , xm) denote the

n Ć— m matrix ā‚xjā’1 ,xj Ā· Ā· Ā· ā‚x2 ,x3 ā‚x1 ,x2 fi (x1) 1ā¤iā¤n, 1ā¤jā¤m . Given a composition of n,

n = m1 + Ā· Ā· Ā· + m , there holds

det Wm1 (a1, . . . , am1 ) Wm2 (am1 +1 , . . . , am1 +m2 ) . . . Wm (am1 +Ā·Ā·Ā·+m ā’1 +1 , . . . , an )

1ā¤i,j,ā¤n

(aj ā’ ai ) . (3.5)

= det (fi (aj ))

1ā¤i,j,ā¤n

m1 +Ā·Ā·Ā·+mkā’1 +1ā¤i<jā¤m1 +Ā·Ā·Ā·+mk

k=1

If we now choose fi (x) := xiā’1 , so that det1ā¤i,j,ā¤n (fi (aj )) is a Vandermonde deter-

minant, then the right-hand side of (3.5) factors completely by (2.1). The ļ¬nal step

to obtain Theorem 20 is to let a1 ā’ X1 , a2 ā’ X1 , . . . , am1 ā’ X1 , am1 +1 ā’ X2 , . . . ,

am1 +m2 ā’ X2 , etc., in (3.5). This does indeed yield (3.1), because

jā’1

1 d

lim . . . lim lim ā‚xjā’1 ,xj Ā· Ā· Ā· ā‚x2 ,x3 ā‚x1 ,x2 g(x1) = g(x),

(j ā’ 1)! dx

xj ā’x x2 ā’x x1 ā’x

as is easily veriļ¬ed.

The Abel-type variation in Theorem 21 follows from Theorem 20 by multiplying

column j in (3.1) by X1 for j = 1, 2, . . . , m1, by X2 1 ā’1 for j = m1 +1, m1 +2, . . . , m2 ,

jā’1 jā’m

etc., and by then using the relation

d d

Xg(X) ā’ g(X)

X g(X) =

dX dX

28 C. KRATTENTHALER

jā’1 iā’1

many times, so that a typical entry Xk (d/dXk )jā’1 Xk in row i and column j of the

iā’1

k-th submatrix is expressed as (Xk (d/dXk ))jā’1 Xk plus a linear combination of terms

(Xk (d/dXk ))s Xk with s < j ā’ 1. Simple column reductions then yield (3.2).

iā’1

It is now not very diļ¬cult to adapt this analysis to derive, for example, q-analogues

of Theorems 20 and 21. The results below do actually contain q-analogues of extensions

of Theorems 20 and 21.

Theorem 23. Let n be a nonnegative integer, and let Am (X) denote the n Ć— m matrix

ļ£«

[C]q X ā’1 [C]q [C ā’ 1]q X ā’2

1

ļ£¬X [C + 1]q [C]q X ā’1

[C + 1]q

ļ£¬2

ļ£¬X [C + 2]q X [C + 2]q [C + 1]q

ļ£¬

ļ£. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

X nā’1 [C + n ā’ 1]q X nā’2 [C + n ā’ 1]q [C + n ā’ 2]q X nā’3

ļ£¶

[C]q Ā· Ā· Ā· [C ā’ m + 2]q X 1ā’m

...

ļ£·

[C + 1]q Ā· Ā· Ā· [C ā’ m + 3]q X 2ā’m

... ļ£·

ļ£·,

[C + 2]q Ā· Ā· Ā· [C ā’ m + 4]q X 3ā’m

... ļ£·

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ļ£ø

. . . [C + n ā’ 1]q Ā· Ā· Ā· [C + n ā’ m + 1]q X nā’m

i.e., any next column is formed by applying the operator X ā’C Dq X C , with Dq denoting

the usual q-derivative, Dq f(X) := (f(qX) ā’ f(X))/(q ā’ 1)X. Given a composition of

n, n = m1 + Ā· Ā· Ā· + m , there holds

det Am1 (X1 ) Am2 (X2 ) . . . Am (X )

1ā¤i,j,ā¤n

mi ā’1 mj ā’1

mi ā’1

(q tā’s Xj ā’ Xi ), (3.6)

= q N1 [j]q !

i=1 j=1 s=0 t=0

1ā¤i<jā¤

where N1 is the quantity

mi

(C + j + m1 + Ā· Ā· Ā· + miā’1 ā’ 1)(mi ā’ j) ā’ ā’ ā’ mj

mi mj mi

mi .

3 2 2

i=1 j=1 1ā¤i<jā¤

To derive (3.6) one would choose strings of geometric sequences for the variables aj

in Lemma 22, i.e., a1 = X1 , a2 = qX1, a3 = q 2X1 , . . . , am1 +1 = X2 , am1 +2 = qX2 , etc.,

and, in addition, use the relation

y C ā‚x,y f(x, y) = ā‚x,y (xC f(x, y)) ā’ (ā‚x,y xC )f(x, y) (3.7)

repeatedly.

A āq-Abel-typeā variation of this result reads as follows.

ADVANCED DETERMINANT CALCULUS 29

Theorem 24. Let n be a nonnegative integer, and let Bm (X) denote the n Ć— m matrix

ļ£« ļ£¶

[C]2 [C]mā’1

1 [C]q ...

q q

ļ£¬X ļ£·

2

[C + 1]mā’1 X

[C + 1]q X [C + 1]q X ...

ļ£¬2 ļ£·

q

ļ£¬X [C + 2]mā’1 X 2 ļ£· ,

[C + 2]q X 2 [C + 2]2 X 2 ...

ļ£¬ ļ£·

q q

ļ£. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ļ£ø

X nā’1 [C + n ā’ 1]q X nā’1 [C + n ā’ 1]2 X nā’1 . . . [C + n ā’ 1]mā’1 X nā’1 q q

i.e., any next column is formed by applying the operator X 1ā’C Dq X C , with Dq denoting

the q-derivative as in Theorem 23. Given a composition of n, n = m1 + Ā· Ā· Ā· + m , there

holds

det Bm1 (X1 ) Bm2 (X2 ) . . . Bm (X )

1ā¤i,j,ā¤n

mi ā’1 mj ā’1

mi ā’1

(mi )

(q tā’s Xj ā’ Xi ), (3.8)

= q N2 2

Xi [j]q !

i=1 j=1 s=0 t=0

1ā¤i<jā¤

where N2 is the quantity

mi

((C + j + m1 + Ā· Ā· Ā· + miā’1 ā’ 1)(mi ā’ j)) ā’ ā’ mj

mj mi

mi .

2 2

i=1 j=1 1ā¤i<jā¤

Yet another generalization of the Vandermonde determinant evaluation is found in

[21]. Multidimensional analogues are contained in [176, Theorem A.7, Eq. (A.14),

Theorem B.8, Eq. (B.11)] and [182, Part I, p. 547].

Extensions of Cauchyā™s double alternant (2.7) can also be found in the literature (see

e.g. [117, 149]). I want to mention here particularly Borchardtā™s variation [17] in which

the (i, j)-entry in Cauchyā™s double alternant is replaced by its square,

ā’ Xj )(Yi ā’ Yj )

1ā¤i<jā¤n (Xi

1 1

det = Per , (3.9)

(Xi ā’ Yj )2 1ā¤i,jā¤n (Xi ā’ Yj )

1ā¤i,jā¤n Xi ā’ Yj

1ā¤i,jā¤n

where Per M denotes the permanent of the matrix M. Thus, there is no closed form

expression such as in (2.7). This may not look that useful. However, most remarkably,

there is a (q-)deformation of this identity which did indeed lead to a āclosed form evalu-

ation,ā thus solving a famous enumeration problem in an unexpected way, the problem

of enumerating alternating sign matrices.10 This q-deformation is equivalent to Izerginā™s

evaluation [74, Eq. (5)] (building on results by Korepin [82]) of the partition function of

the six-vertex model under certain boundary conditions (see also [97, Theorem 8] and

[83, Ch. VII, (10.1)/(10.2)]).

An alternating sign matrix is a square matrix with entries 0, 1, ā’1, with all row and column

10

sums equal to 1, and such that, on disregarding the 0s, in each row and column the 1s and (ā’1)s

alternate. Alternating sign matrix are currently the most fascinating, and most mysterious, objects in

enumerative combinatorics. The reader is referred to [18, 19, 111, 148, 97, 198, 199] for more detailed

material. Incidentally, the ābirthā of alternating sign matrices came through ā” determinants, see

[150].

30 C. KRATTENTHALER

Theorem 25. For any nonnegative integer n there holds

ā’ Xj )(Yi ā’ Yj )

1ā¤i<jā¤n (Xi

1

det =

(Xi ā’ Yj )(qXi ā’ Yj ) 1ā¤i,jā¤n (Xi ā’ Yj )(qXi ā’ Yj )

1ā¤i,jā¤n

n

N i (A)

Ni (A)

Ć— (1 ā’ q) (Ī±i,j Xi ā’ Yj ), (3.10)

2N (A)

Xi Yi

i=1

A i,j such that Aij =0

where the sum is over all n Ć— n alternating sign matrices A = (Aij )1ā¤i,jā¤n , N(A) is

the number of (ā’1)s in A, Ni (A) (respectively N i (A)) is the number of (ā’1)s in the

i-th row (respectively column) of A, and Ī±ij = q if j Aik = i Akj , and Ī±ij = 1

k=1 k=1

otherwise.

Clearly, equation (3.9) results immediately from (3.10) by setting q = 1. Roughly,

Kuperbergā™s solution [97] of the enumeration of alternating sign matrices consisted of

suitably specializing the xi ā™s, yi ā™s and q in (3.10), so that each summand on the right-

hand side would reduce to the same quantity, and, thus, the sum would basically count

n Ć— n alternating sign matrices, and in evaluating the left-hand side determinant for

that special choice of the xi ā™s, yi ā™s and q. The resulting number of n Ć— n alternating

sign matrices is given in (A.1) in the Appendix. (The ļ¬rst, very diļ¬erent, solution

is due to Zeilberger [198].) Subsequently, Zeilberger [199] improved on Kuperbergā™s

approach and succeeded in proving the reļ¬ned alternating sign matrix conjecture from

[111, Conj. 2]. For a diļ¬erent expansion of the determinant of Izergin, in terms of Schur

functions, and a variation, see [101, Theorem q, Theorem Ī³].

Next we turn to typical applications of Lemma 3. They are listed in the following

theorem.

Theorem 26. Let n be a nonnegative integer, and let L1, L2 , . . . , Ln and A, B be inde-

terminates. Then there hold

1ā¤i<jā¤n [Li ā’ Lj ]q

n

i=1 [Li + A + 1]q !

Li + A + j n

(iā’1)(Li +i)

det = q i=1 ,

[A + 1 ā’ i]q !

n n

Li + j [Li + n]q !

1ā¤i,jā¤n

q i=1 i=1

(3.11)

and

1ā¤i<jā¤n [Li ā’ Lj ]q i=1 [A + i ā’ 1]q !

n

A n

jLi iLi

det q =q , (3.12)

i=1

[A ā’ Li ā’ 1]q !

n n

Li + j i=1 [Li + n]q !

1ā¤i,jā¤n

q i=1

and

BLi + A

det

Li + j

1ā¤i,jā¤n

1ā¤i<jā¤n (Li ā’ Lj )

n n

(BLi + A)!

(A ā’ Bi + 1)iā’1 , (3.13)

=

((B ā’ 1)Li + A ā’ 1)!

n

i=1 (Li + n)! i=1 i=1

and

n

(A + BLi )jā’1 (A + Bi)iā’1

(Lj ā’ Li ).

det = (3.14)

(j ā’ Li )! (n ā’ Li )!

1ā¤i,jā¤n

i=1 1ā¤i<jā¤n

ADVANCED DETERMINANT CALCULUS 31

(For derivations of (3.11) and (3.12) using Lemma 3 see the proofs of Theorems 6.5

and 6.6 in [85]. For a derivation of (3.13) using Lemma 3 see the proof of Theorem 5

in [86].)

Actually, the evaluations (3.11) and (3.12) are equivalent. This is seen by observing

that

ā’A ā’ 1

Li + A + j Li j

= (ā’1)Li +j q ( 2 )+(2)+jLi +(A+1)(Li +j) .

Li + j Li + j q

q

Hence, replacement of A by ā’A ā’ 1 in (3.11) leads to (3.12) after little manipulation.

The determinant evaluations (3.11) and (3.12), and special cases thereof, are redis-

covered and reproved in the literature over and over. (This phenomenon will probably

persist.) To the best of my knowledge, the evaluation (3.11) appeared in print explicitly

for the ļ¬rst time in [22], although it was (implicitly) known earlier to people in group

representation theory, as it also results from the principal specialization (i.e., set xi = q i ,

i = 1, 2, . . . , N) of a Schur function of arbitrary shape, by comparing the Jacobiā“Trudi

identity with the bideterminantal form (Weyl character formula) of the Schur function

(cf. [105, Ch. I, (3.4), Ex. 3 in Sec. 2, Ex. 1 in Sec. 3]; the determinants arising in the

bideterminantal form are Vandermonde determinants and therefore easily evaluated).

The main applications of (3.11)ā“(3.13) are in the enumeration of tableaux, plane par-

titions and rhombus tilings. For example, the hook-content formula [163, Theorem 15.3]

for tableaux of a given shape with bounded entries follows immediately from the the-

ory of nonintersecting lattice paths (cf. [57, Cor. 2] and [169, Theorem 1.2]) and the

determinant evaluation (3.11) (see [57, Theorem 14] and [85, proof of Theorem 6.5]).

MacMahonā™s ābox formulaā [106, Sec. 429; proof in Sec. 494] for the generating function

of plane partitions which are contained inside a given box follows from nonintersecting

lattice paths and the determinant evaluation (3.12) (see [57, Theorem 15] and [85, proof

of Theorem 6.6]). The q = 1 special case of the determinant which is relevant here is

the one in (1.2) (which is the one which was evaluated as an illustration in Section 2.2).

To the best of my knowledge, the evaluation (3.13) is due to Proctor [133] who used

it for enumerating plane partitions of staircase shape (see also [86]). The determinant

evaluation (3.14) can be used to give closed form expressions in the enumeration of Ī»-

parking functions (an extension of the notion of k-parking functions such as in [167]), if

one starts with determinantal expressions due to Gessel (private communication). Fur-

ther applications of (3.11), in the domain of multiple (basic) hypergeometric series, are

found in [63]. Applications of these determinant evaluations in statistics are contained

in [66] and [168].

It was pointed out in [34] that plane partitions in a given box are in bijection with

rhombus tilings of a āsemiregularā hexagon. Therefore, the determinant (1.2) counts

as well rhombus tilings in a hexagon with side lengths a, b, n, a, b, n. In this regard,

generalizations of the evaluation of this determinant, and of a special case of (3.13),

appear in [25] and [27]. The theme of these papers is to enumerate rhombus tilings of

a hexagon with triangular holes.

The next theorem provides a typical application of Lemma 4. For a derivation of this

determinant evaluation using this lemma see [87, proofs of Theorems 8 and 9].

32 C. KRATTENTHALER

Theorem 27. Let n be a nonnegative integer, and let L1 , L2 , . . . , Ln and A be indeter-

minates. Then there holds

Li + A ā’ j

q jLi

det

Li + j

1ā¤i,jā¤n

q

n

[Li + A ā’ n]q !

n

[Li ā’ Lj ]q [Li + Lj + A + 1]q . (3.15)

iLi

=q i=1

[Li + n]q ! [A ā’ 2i]q ! 1ā¤i<jā¤n

i=1

This result was used to compute generating functions for shifted plane partitions of

trapezoidal shape (see [87, Theorems 8 and 9], [134, Prop. 4.1] and [135, Theorem 1]).

Now we turn to typical applications of Lemma 5, given in Theorems 28ā“31 below.

All of them can be derived in just the same way as we evaluated the determinant (1.2)

in Section 2.2 (the only diļ¬erence being that Lemma 5 is invoked instead of Lemma 3).

The ļ¬rst application is the evaluation of a determinant whose entries are a product

of two q-binomial coeļ¬cients.

Theorem 28. Let n be a nonnegative integer, and let L1, L2 , . . . , Ln and A, B be inde-

terminates. Then there holds

Li + A ā’ j

Li + j

Ā·

det

B B

1ā¤i,jā¤n

q q

( )+2(n+1)

n

i=1 (iā’1)Li ā’B 2

n

[Li ā’ Lj ]q [Li + Lj + A ā’ B + 1]q

=q 3

1ā¤i<jā¤n

n

[Li + 1]q ! [Li + A ā’ n]q ! [A ā’ 2i ā’ 1]q !

Ć— . (3.16)

[Li ā’ B + n]q ! [Li + A ā’ B ā’ 1]q ! [A ā’ i ā’ n ā’ 1]q ! [B + i ā’ n]q ! [B]q !

i=1

As is not diļ¬cult to verify, this determinant evaluation contains (3.11), (3.12), as

well as (3.15) as special, respectively limiting cases.

This determinant evaluation found applications in basic hypergeometric functions

theory. In [191, Sec. 3], Wilson used a special case to construct biorthogonal rational

functions. On the other hand, Schlosser applied it in [157] to ļ¬nd several new summation

theorems for multidimensional basic hypergeometric series.

In fact, as Joris Van der Jeugt pointed out to me, there is a generalization of Theo-

rem 28 of the following form (which can be also proved by means of Lemma 5).

ADVANCED DETERMINANT CALCULUS 33

Theorem 29. Let n be a nonnegative integer, and let X0 , X1 , . . . , Xnā’1 , Y0 , Y1 , . . . ,

Ynā’1 , A and B be indeterminates. Then there holds

ļ£« ļ£¶

Yj + A ā’ Xi

Xi + Yj

ļ£¬ ļ£·

j j

ļ£¬ qļ£·

q

det ļ£¬ ļ£·

0ā¤i,jā¤nā’1 ļ£ Xi + B A + B ā’ Xi ļ£ø

j j

q q

n nā’1

i(Xi +Yi ā’Aā’2B)

= q 2(3 )+ [Xi ā’ Xj ]q [Xi + Xj ā’ A]q

i=0

0ā¤i<jā¤nā’1

nā’1

(q Bā’Yi ā’i+1 )i (q Yi +A+B+2ā’2i )i

Ć— . (3.17)

(q Xiā’Aā’B )nā’1 (q Xi+Bā’n+2 )nā’1

i=0

As another application of Lemma 5 we list two evaluations of determinants (see below)

where the entries are, up to some powers of q, a diļ¬erence of two q-binomial coeļ¬cients.

A proof of the ļ¬rst evaluation which uses Lemma 5 can be found in [88, proof of

Theorem 7], a proof of the second evaluation using Lemma 5 can be found in [155,

Ch. VI, Ā§3]. Once more, the second evaluation was always (implicitly) known to people

in group representation theory, as it also results from a principal specialization (set

xi = q iā’1/2, i = 1, 2, . . . ) of a symplectic character of arbitrary shape, by comparing the

symplectic dual Jacobiā“Trudi identity with the bideterminantal form (Weyl character

formula) of the symplectic character (cf. [52, Cor. 24.24 and (24.18)]; the determinants

arising in the bideterminantal form are easily evaluated by means of (2.4)).

Theorem 30. Let n be a nonnegative integer, and let L1 , L2 , . . . , Ln and A be indeter-

minates. Then there hold

A A

q j(Lj ā’Li ) ā’ q j(2Li+Aā’1)

det

j ā’ Li ā’j ā’ Li + 1

1ā¤i,jā¤n

q q

n

[A + 2i ā’ 2]q !

[Lj ā’ Li ]q [Li + Lj + A ā’ 1]q (3.18)

=

[n ā’ Li ]q ! [A + n ā’ 1 + Li ]q ! 1ā¤i<jā¤n

i=1 1ā¤iā¤jā¤n

and

A A

q j(Lj ā’Li ) ā’ q j(2Li+A)

det

j ā’ Li ā’j ā’ Li

1ā¤i,jā¤n

q q

n

[A + 2i ā’ 1]q !

[Lj ā’ Li ]q

= [Li + Lj + A]q . (3.19)

[n ā’ Li ]q ! [A + n + Li ]q ! 1ā¤i<jā¤n

i=1 1ā¤iā¤jā¤n

A special case of (3.19) was the second determinant evaluation which Andrews needed

in [4, (1.4)] in order to prove the MacMahon Conjecture (since then, ex-Conjecture)

about the q-enumeration of symmetric plane partitions. Of course, Andrewsā™ evaluation

proceeded by LU-factorization, while Schlosser [155, Ch. VI, Ā§3] simpliļ¬ed Andrewsā™

proof signiļ¬cantly by making use of Lemma 5. The determinant evaluation (3.18)

34 C. KRATTENTHALER

was used in [88] in the proof of reļ¬nements of the MacMahon (ex-)Conjecture and the

Benderā“Knuth (ex-)Conjecture. (The latter makes an assertion about the generating

function for tableaux with bounded entries and a bounded number of columns. The

ļ¬rst proof is due to Gordon [59], the ļ¬rst published proof [3] is due to Andrews.)

Next, in the theorem below, we list two very similar determinant evaluations. This

time, the entries of the determinants are, up to some powers of q, a sum of two q-

binomial coeļ¬cients. A proof of the ļ¬rst evaluation which uses Lemma 5 can be found

in [155, Ch. VI, Ā§3]. A proof of the second evaluation can be established analogously.

Again, the second evaluation was always (implicitly) known to people in group represen-

tation theory, as it also results from a principal specialization (set xi = q i, i = 1, 2, . . . )

of an odd orthogonal character of arbitrary shape, by comparing the orthogonal dual

Jacobiā“Trudi identity with the bideterminantal form (Weyl character formula) of the

orthogonal character (cf. [52, Cor. 24.35 and (24.28)]; the determinants arising in the

bideterminantal form are easily evaluated by means of (2.3)).

Theorem 31. Let n be a nonnegative integer, and let L1 , L2 , . . . , Ln and A be indeter-

minates. Then there hold

A A

q (jā’1/2)(Lj ā’Li ) + q (jā’1/2)(2Li+Aā’1)

det

j ā’ Li ā’j ā’ Li + 1

1ā¤i,jā¤n

q q

n

[A + 2i ā’ 1]q !

(1 + q Li +A/2ā’1/2)

=

(1 + q i+A/2ā’1/2) [n ā’ Li ]q ! [A + n + Li ā’ 1]q !

i=1

Ć— [Lj ā’ Li ]q [Li + Lj + A ā’ 1]q (3.20)

1ā¤i<jā¤n

and

A A

q (jā’1/2)(Lj ā’Li ) + q (jā’1/2)(2Li+Aā’2)

det

j ā’ Li ā’j ā’ Li + 2

1ā¤i,jā¤n

q q

n

[A + 2i ā’ 2]q !

n Li+A/2ā’1

i=1 (1 + q )

=

[n ā’ Li ]q ! [A + n + Li ā’ 2]q !

n i+A/2ā’1 )

i=2 (1 + q i=1

Ć— [Lj ā’ Li ]q [Li + Lj + A ā’ 2]q . (3.21)

1ā¤i<jā¤n

A special case of (3.20) was the ļ¬rst determinant evaluation which Andrews needed

in [4, (1.3)] in order to prove the MacMahon Conjecture on symmetric plane parti-

tions. Again, Andrewsā™ evaluation proceeded by LU-factorization, while Schlosser [155,

Ch. VI, Ā§3] simpliļ¬ed Andrewsā™ proof signiļ¬cantly by making use of Lemma 5.

Now we come to determinants which belong to a diļ¬erent category what regards

diļ¬culty of evaluation, as it is not possible to introduce more parameters in a substantial

way.

The ļ¬rst determinant evaluation in this category that we list here is a determinant

evaluation due to Andrews [5, 6]. It solved, at the same time, Macdonaldā™s problem of

ADVANCED DETERMINANT CALCULUS 35

enumerating cyclically symmetric plane partitions and Andrewsā™ own conjecture about

the enumeration of descending plane partitions.

Theorem 32. Let Āµ be an indeterminate. For nonnegative integers n there holds

2Āµ + i + j

det Ī“ij +

j

0ā¤i,jā¤nā’1

ļ£± nā’2

ļ£“ n/2

ļ£“2

ļ£“

ļ£“ (Āµ + i/2 + 1) (i+3)/4

ļ£“

ļ£“

ļ£“

ļ£“ i=1

ļ£“

ļ£“

i=1 Āµ + 2 ā’ 2 + 2 i/2 ā’1 Āµ + 2 ā’ 2 +

n/2

ļ£“ 3n 3i 3 3n 3i 3

ļ£“

ļ£“Ć— 2 i/2

ļ£“ if n is even,

ļ£²

i=1 (2i ā’ 1)!! (2i + 1)!!

n/2ā’1

=

ļ£“ n/2 nā’2

ļ£“

ļ£“2

ļ£“ (Āµ + i/2 + 1) (i+3)/4

ļ£“

ļ£“

ļ£“

ļ£“

ļ£“ i=1

ļ£“

ļ£“ Āµ + 3n ā’ 3iā’1 + 1 (iā’1)/2 Āµ + 3n ā’

(nā’1)/2 3i

ļ£“

ļ£“Ć— i=1

ļ£“ 2 2 2 2 i/2

ļ£³ if n is odd.

(2i ā’ 1)!!2

(nā’1)/2

i=1

(3.22)

The specializations of this determinant evaluation which are of relevance for the

enumeration of cyclically symmetric plane partitions and descending plane partitions

are the cases Āµ = 0 and Āµ = 1, respectively. In these cases, Macdonald, respectively

Andrews, actually had conjectures about q-enumeration. These were proved by Mills,

Robbins and Rumsey [110]. Their theorem which solves the q-enumeration of cyclically

symmetric plane partitions is the following.

Theorem 33. For nonnegative integers n there holds

n

1 ā’ q 3iā’1 1 ā’ q 3(n+i+jā’1)

i+j

3i+1

det Ī“ij + q = . (3.23)

1 ā’ q 3iā’2 1ā¤iā¤jā¤n 1 ā’ q 3(2i+jā’1)

j

0ā¤i,jā¤nā’1

q3 i=1

The theorem by Mills, Robbins and Rumsey in [110] which concerns the enumeration

of descending plane partitions is the subject of the next theorem.

Theorem 34. For nonnegative integers n there holds

1 ā’ q n+i+j

i+j +2

i+2

det Ī“ij + q = . (3.24)

1 ā’ q 2i+jā’1

j

0ā¤i,jā¤nā’1

q 1ā¤iā¤jā¤n+1

It is somehow annoying that so far nobody was able to come up with a full q-analogue

of the Andrews determinant (3.22) (i.e., not just in the cases Āµ = 0 and Āµ = 1). This

issue is already addressed in [6, Sec. 3]. In particular, it is shown there that the result

for a natural q-enumeration of a parametric family of descending plane partitions does

not factor nicely in general, and thus does not lead to a q-analogue of (3.22). Yet, such

36 C. KRATTENTHALER

a q-analogue should exist. Probably the binomial coeļ¬cient in (3.22) has to be replaced

by something more complicated than just a q-binomial times some power of q.

On the other hand, there are surprising variations of the Andrews determinant (3.22),

discovered by Douglas Zare. These can be interpreted as certain weighted enumerations

of cyclically symmetric plane partitions and of rhombus tilings of a hexagon with a

triangular hole (see [27]).

Theorem 35. Let Āµ be an indeterminate. For nonnegative integers n there holds

2Āµ + i + j

ā’Ī“ij +

det

j

0ā¤i,jā¤nā’1

0, if n is odd,

= (3.25)

i!2 (Āµ+i)!2 (Āµ+3i+1)!2 (2Āµ+3i+1)!2

n/2ā’1

(ā’1)n/2 , if n is even.

(2i)! (2i+1)! (Āµ+2i)!2 (Āµ+2i+1)!2 (2Āµ+2i)! (2Āµ+2i+1)!

i=0

If Ļ is a primitive 3rd root of unity, then for nonnegative integers n there holds

(1 + Ļ)n2 n/2

2Āµ + i + j

det ĻĪ“ij + =

(2i ā’ 1)!! (2i ā’ 1)!!

n/2 (nā’1)/2

j

0ā¤i,jā¤nā’1

i=1 i=1

Ć— (Āµ + 3i + 1) (Āµ + 3i + 3)

(nā’4i)/2 (nā’4iā’3)/2

iā„0

Ā· Āµ+nā’i+ Āµ+nā’iā’

1 1

, (3.26)

2 2

(nā’4iā’1)/2 (nā’4iā’2)/2

where, in abuse of notation, by Ī± we mean the usual ļ¬‚oor function if Ī± ā„ 0, however,

if Ī± < 0 then Ī± must be read as 0, so that the product over i in (3.26) is indeed a

ļ¬nite product.

If Ļ is a primitive 6th root of unity, then for nonnegative integers n there holds

2 n/2

(1 + Ļ)n

2Āµ + i + j 3

det ĻĪ“ij + =

(2i ā’ 1)!! (2i ā’ 1)!!

n/2 (nā’1)/2

j

0ā¤i,jā¤nā’1

i=1 i=1

Ć— 3 5

Āµ + 3i + Āµ + 3i +

2 2

(nā’4iā’1)/2 (nā’4iā’2)/2

iā„0

Ā· (Āµ + n ā’ i) (Āµ + n ā’ i) , (3.27)

(nā’4i)/2 (nā’4iā’3)/2

where again, in abuse of notation, by Ī± we mean the usual ļ¬‚oor function if Ī± ā„ 0,

however, if Ī± < 0 then Ī± must be read as 0, so that the product over i in (3.27) is

indeed a ļ¬nite product.

There are no really simple proofs of Theorems 32ā“35. Let me just address the issue

of proofs of the evaluation of the Andrews determinant, Theorem 32. The only direct

proof of Theorem 32 is the original proof of Andrews [5], who worked out the LU-

factorization of the determinant. Today one agrees that the āeasiestā way of evaluating

the determinant (3.22) is by ļ¬rst employing a magniļ¬cent factorization theorem [112,

Theorem 5] due to Mills, Robbins and Rumsey, and then evaluating each of the two

resulting determinants. For these, for some reason, more elementary evaluations exist

(see in particular [10] for such a derivation). What I state below is a (straightforward)

generalization of this factorization theorem from [92, Lemma 2].

ADVANCED DETERMINANT CALCULUS 37

Theorem 36. Let Zn (x; Āµ, Ī½) be deļ¬ned by

nā’1 nā’1

j ā’ k + Āµ ā’ 1 kā’t

i+Āµ k+Ī½

Zn (x; Āµ, Ī½) := det Ī“ij + x ,

kā’t jā’k

t

0ā¤i,jā¤nā’1

t=0 k=0

let Tn (x; Āµ, Ī½) be deļ¬ned by

2j

i+Āµ j+Ī½

x2jā’t ,

Tn (x; Āµ, Ī½) := det

tā’i 2j ā’ t

0ā¤i,jā¤nā’1

t=i

and let Rn (x; Āµ, Ī½) be deļ¬ned by

2j+1

i+Āµ i+Āµ+1

Rn (x; Āµ, Ī½) := det +

tā’iā’1 tā’i

0ā¤i,jā¤nā’1

t=i

j+Ī½ j+Ī½+1

Ā· x2j+1ā’t .

+

2j + 1 ā’ t 2j + 1 ā’ t

Then for all positive integers n there hold

Z2n(x; Āµ, Ī½) = Tn (x; Āµ, Ī½/2) Rn (x; Āµ, Ī½/2) (3.28)

and

Z2nā’1 (x; Āµ, Ī½) = 2 Tn(x; Āµ, Ī½/2) Rnā’1 (x; Āµ, Ī½/2). (3.29)

The reader should observe that Zn (1; Āµ, 0) is identical with the determinant in (3.22),

as the sums in the entries simplify by means of Chuā“Vandermonde summation (see e.g.

[62, Sec. 5.1, (5.27)]). However, also the entries in the determinants Tn (1; Āµ, 0) and

Rn(1; Āµ, 0) simplify. The respective evaluations read as follows (see [112, Theorem 7]

and [9, (5.2)/(5.3)]).

Theorem 37. Let Āµ be an indeterminate. For nonnegative integers n there holds

Āµ+i+j

det

2i ā’ j

0ā¤i,jā¤nā’1

ā’Āµ ā’ 3n + i + 3

nā’1 (Āµ + i + 1) (i+1)/2

nā’1 2

2( ) i/2

= (ā’1)Ļ(nā”3 mod 4)

, (3.30)

2

(i)i

i=1

where Ļ(A) = 1 if A is true and Ļ(A) = 0 otherwise, and

Āµ+i+j Āµ+i+j +2

det +2

2i ā’ j 2i ā’ j + 1

0ā¤i,jā¤nā’1

n

(Āµ + 3n ā’ 3iā’1 + 1 )

(Āµ + i) i/2 (i+1)/2

n 2 2

=2 . (3.31)

(2i ā’ 1)!!

i=1

38 C. KRATTENTHALER

The reader should notice that the determinant in (3.30) is the third determinant from

the Introduction, (1.3). Originally, in [112, Theorem 7], Mills, Robbins and Rumsey

proved (3.30) by applying their factorization theorem (Theorem 36) the other way

round, relying on Andrewsā™ Theorem 32. However, in the meantime there exist short

direct proofs of (3.30), see [10, 91, 129], either by LU-factorization, or by āidentiļ¬cation

of factorsā. A proof based on the determinant evaluation (3.35) and some combinatorial

considerations is given in [29, Remark 4.4], see the remarks after Theorem 40. As shown

in [9, 10], the determinant (3.31) can easily be transformed into a special case of the

determinant in (3.35) (whose evaluation is easily proved using condensation, see the

corresponding remarks there). Altogether, this gives an alternative, and simpler, proof

of Theorem 32.

Mills, Robbins and Rumsey needed the evaluation of (3.30) because it allowed them

to prove the (at that time) conjectured enumeration of cyclically symmetric transpose-

complementary plane partitions (see [112]). The unspecialized determinants Zn (x; Āµ, Ī½)

and Tn (x; Āµ, Ī½) have combinatorial meanings as well (see [110, Sec. 4], respectively

[92, Sec. 3]), as the weighted enumeration of certain descending plane partitions and

triangularly shaped plane partitions.

It must be mentioned that the determinants Zn (x; Āµ, Ī½), Tn (x; Āµ, Ī½), Rn (x; Āµ, Ī½) do

also factor nicely for x = 2. This was proved by Andrews [7] using LU-factorization,

thus conļ¬rming a conjecture by Mills, Robbins and Rumsey (see [92] for an alternative

proof by āidentiļ¬cation of factorsā).

It was already mentioned in Section 2.8 that there is a general theorem by Goulden

and Jackson [61, Theorem 2.1] (see Lemma 19 and the remarks thereafter) which,

given the evaluation (3.30), immediately implies a generalization containing one more

parameter. (This property of the determinant (3.30) is called by Goulden and Jackson

the averaging property.) The resulting determinant evaluation had been earlier found

by Andrews and Burge [9, Theorem 1]. They derived it by showing that it can be

obtained by multiplying the matrix underlying the determinant (3.30) by a suitable

triangular matrix.

Theorem 38. Let x and y be indeterminates. For nonnegative integers n there holds

x+i+j y+i+j

det +

2i ā’ j 2i ā’ j

0ā¤i,jā¤nā’1

ā’ x+y ā’ 3n + i +

nā’1 x+y + i + 1 3

n 2 2 2

= (ā’1)Ļ(nā”3 mod 4)2( 2 )+1

(i+1)/2 i/2

, (3.32)

(i)i

i=1

where Ļ(A) = 1 if A is true and Ļ(A) = 0 otherwise.

(The evaluation (3.32) does indeed reduce to (3.30) by setting x = y.)

The above described procedure of Andrews and Burge to multiply a matrix, whose

determinant is known, by an appropriate triangular matrix, and thus obtain a new

determinant evaluation, was systematically exploited by Chu [23]. He derives numerous

variations of (3.32), (3.31), and special cases of (3.13). We content ourselves with

displaying two typical identities from [23, (3.1a), (3.5a)], just enough to get an idea of

the character of these.

ADVANCED DETERMINANT CALCULUS 39

Theorem 39. Let x0, x1 , . . . , xnā’1 and c be indeterminates. For nonnegative integers

n there hold

c ā’ xi + i + j

c + xi + i + j

det +

2i ā’ j 2i ā’ j

0ā¤i,jā¤nā’1

ā’c ā’ 3n + i + 3

nā’1 (c + i + 1)

(i+1)/2

n 2

Ļ(nā”3 mod 4) ( 2 )+1 i/2

= (ā’1) 2 (3.33)

(i)i

i=1

and

(2i ā’ j) + (2c + 3j + 1)(2c + 3j ā’ 1) c + i + j + 1

2

det

2i ā’ j

(c + i + j + 1 )(c + i + j ā’ 1 )

0ā¤i,jā¤nā’1

2 2

(ā’c ā’ 3n + i + 2)

1

nā’1 c+i+ i/2

n+1 2

2( )+1 (i+1)/2

Ļ(nā”3 mod 4)

= (ā’1) , (3.34)

2

(i)i

i=1

where Ļ(A) = 1 if A is true and Ļ(A) = 0 otherwise.

The next determinant (to be precise, the special case y = 0), whose evaluation is

stated in the theorem below, seems to be closely related to the Millsā“Robbinsā“Rumsey

determinant (3.30), although it is in fact a lot easier to evaluate. Indications that

the evaluation (3.30) is much deeper than the following evaluation are, ļ¬rst, that it

does not seem to be possible to introduce a second parameter into the Millsā“Robbinsā“

Rumsey determinant (3.30) in a similar way, and, second, the much more irregular form

of the right-hand side of (3.30) (it contains many ļ¬‚oor functions!), as opposed to the

right-hand side of (3.35).

Theorem 40. Let x, y, n be nonnegative integers. Then there holds

(x + y + i + j ā’ 1)!

det

(x + 2i ā’ j)! (y + 2j ā’ i)!

0ā¤i,jā¤nā’1

nā’1

i! (x + y + i ā’ 1)! (2x + y + 2i)i (x + 2y + 2i)i

= . (3.35)

(x + 2i)! (y + 2i)!

i=0

This determinant evaluation is due to the author, who proved it in [90, (5.3)] as an

aside to the (much more diļ¬cult) determinant evaluations which were needed there to

settle a conjecture by Robbins and Zeilberger about a generalization of the enumeration

of totally symmetric self-complementary plane partitions. (These are the determinant

evaluations of Theorems 43 and 45 below.) It was proved there by āidentiļ¬cation of

factorsā. However, Amdeberhan [2] observed that it can be easily proved by āconden-

sationā.

Originally there was no application for (3.35). However, not much later, Ciucu [29]

found not just one application. He observed that if the determinant evaluation (3.35)

is suitably combined with his beautiful Matchings Factorization Theorem [26, Theo-

rem 1.2] (and some combinatorial considerations), then not only does one obtain simple

proofs for the evaluation of the Andrews determinant (3.22) and the Millsā“Robbinsā“

Rumsey determinant (3.30), but also simple proofs for the enumeration of four diļ¬erent

40 C. KRATTENTHALER

symmetry classes of plane partitions, cyclically symmetric plane partitions, cyclically

symmetric self-complementary plane partitions (ļ¬rst proved by Kuperberg [96]), cycli-

cally symmetric transpose-complementary plane partitions (ļ¬rst proved by Mills, Rob-

bins and Rumsey [112]), and totally symmetric self-complementary plane partitions (ļ¬rst

proved by Andrews [8]).

A q-analogue of the previous determinant evaluation is contained in [89, Theorem 1].

Again, Amdeberhan [2] observed that it can be easily proved by means of ācondensa-

tionā.

Theorem 41. Let x, y, n be nonnegative integers. Then there holds

q ā’2ij

(q; q)x+y+i+jā’1

det

(q; q)x+2iā’j (q; q)y+2jā’i (ā’q x+y+1 ; q)i+j

0ā¤i,jā¤nā’1

nā’1 2

; q 2)i (q; q)x+y+iā’1 (q 2x+y+2i ; q)i (q x+2y+2i ; q)i

ā’2i2 (q

= q . (3.36)

(q; q)x+2i (q; q)y+2i (ā’q x+y+1 ; q)nā’1+i

i=0

The reader should observe that this is not a straightforward q-analogue of (3.35) as it

does contain the terms (ā’q x+y+1 ; q)i+j in the determinant, respectively (ā’q x+y+1 ; q)nā’1+i

in the denominator of the right-hand side product, which can be cleared only if q = 1.

A similar determinant evaluation, with some overlap with (3.36), was found by An-

drews and Stanton [10, Theorem 8] by making use of LU-factorization, in their āĀ“tudeā e

on the Andrews and the Millsā“Robbinsā“Rumsey determinant.

Theorem 42. Let x and E be indeterminates and n be a nonnegative integer. Then

there holds

(E/xq i ; q 2)iā’j (q/Exq i; q 2)iā’j (1/x2 q 2+4i; q 2)iā’j

det

(q; q)2i+1ā’j (1/Exq 2i; q)iā’j (E/xq 1+2i; q)iā’j

0ā¤i,jā¤nā’1

nā’1

(x2q 2i+1; q)i (xq 3+i/E; q 2 )i (Exq 2+i; q 2)i

= . (3.37)

(x2q 2i+2; q 2)i (q; q 2)i+1 (Exq 1+i ; q)i (xq 2+i /E; q)i

i=0

The next group of determinants is (with one exception) from [90]. These determi-

nants were needed in the proof of a conjecture by Robbins and Zeilberger about a

generalization of the enumeration of totally symmetric self-complementary plane parti-

tions.

Theorem 43. Let x, y, n be nonnegative integers. Then

(x + y + i + j ā’ 1)! (y ā’ x + 3j ā’ 3i)

det

(x + 2i ā’ j + 1)! (y + 2j ā’ i + 1)!

0ā¤i,jā¤nā’1

nā’1

i! (x + y + i ā’ 1)! (2x + y + 2i + 1)i (x + 2y + 2i + 1)i

=

(x + 2i + 1)! (y + 2i + 1)!

i=0

n

n

Ā· (ā’1)k (x)k (y)nā’k . (3.38)

k

k=0

ADVANCED DETERMINANT CALCULUS 41

This is Theorem 8 from [90]. A q-analogue, provided in [89, Theorem 2], is the

following theorem.

Theorem 44. Let x, y, n be nonnegative integers. Then there holds

(q; q)x+y+i+jā’1 (1 ā’ q y+2jā’i ā’ q y+2jā’i+1 + q x+y+i+j+1 )

det

(q; q)x+2iā’j+1 (q; q)y+2jā’i+1

0ā¤i,jā¤nā’1

q ā’2ij

Ā·

(ā’q x+y+2 ; q)i+j

nā’1 2

; q 2)i (q; q)x+y+iā’1 (q 2x+y+2i+1 ; q)i (q x+2y+2i+1 ; q)i

ā’2i2 (q

= q

(q; q)x+2i+1 (q; q)y+2i+1 (ā’q x+y+2 ; q)nā’1+i

i=0

n

n

Ć— (ā’1)k q nk q yk (q x; q)k (q y ; q)nā’k . (3.39)

k q

k=0

Once more, Amdeberhan observed that, in principle, Theorem 43 as well as The-

orem 44 could be proved by means of ācondensationā. However, as of now, nobody

provided a proof of the double sum identities which would establish (2.16) in these

cases.

We continue with Theorems 2 and Corollary 3 from [90].

ā¤ n. Under the convention

Theorem 45. Let x, m, n be nonnegative integers with m

that sums are interpreted by

ļ£±

ļ£“ B

ļ£² r=A+1 Expr(r) A<B

B

Expr(r) = 0 A=B

ļ£“

ļ£³ā’ A

r=A+1

r=B+1 Expr(r) A > B,

there holds

2x + m + i + j

det

r

0ā¤i,jā¤nā’1

x+2iā’j<rā¤x+m+2jā’i

nā’1

(2x + m + i)! (3x + m + 2i + 2)i (3x + 2m + 2i + 2)i

=

(x + 2i)! (x + m + 2i)!

i=1

n/2 ā’1

(2x + m)!

Ć— Ā· (2x + 2 m/2 + 2i + 1) Ā· P1 (x; m, n), (3.40)

(x + m/2 )! (x + m)! i=0

where P1 (x; m, n) is a polynomial in x of degree ā¤ m/2 .

In particular, for m = 0 the determinant equals

ļ£±

ļ£“ n/2ā’1

ļ£“nā’1

ļ£“ (2x + 2i + 1)

ļ£² i! (2x + i)! (3x + 2i + 2)2 i=0

i

n even (3.41)

(n ā’ 1)!!

ļ£“ i=0 (x + 2i)!2

ļ£“

ļ£“

ļ£³

0 n odd,

42 C. KRATTENTHALER

for m = 1, n ā„ 1, it equals

n/2 ā’1

(2x + 2i + 3)

nā’1

i! (2x + i + 1)! (3x + 2i + 3)i (3x + 2i + 4)i i=0

, (3.42)

(2 n/2 ā’ 1)!!

(x + 2i)! (x + 2i + 1)!

i=0

for m = 2, n ā„ 2, it equals

n/2 ā’1

(2x + 2i + 3)

nā’1

i! (2x + i + 2)! (3x + 2i + 4)i (3x + 2i + 6)i i=0

(2 n/2 ā’ 1)!!

(x + 2i)! (x + 2i + 2)!

i=0

1 (x + n + 1) n even

Ć— Ā· (3.43)

(x + 1) (2x + n + 2) n odd,

for m = 3, n ā„ 3, it equals

n/2 ā’1

(2x + 2i + 5)

nā’1

i! (2x + i + 3)! (3x + 2i + 5)i (3x + 2i + 8)i i=0

(2 n/2 ā’ 1)!!

(x + 2i)! (x + 2i + 3)!

i=0

1 (x + 2n + 1) n even

Ć— Ā· (3.44)

(x + 1) (3x + 2n + 5) n odd,

and for m = 4, n ā„ 4, it equals

n/2 ā’1

(2x + 2i + 5)

nā’1

i! (2x + i + 4)! (3x + 2i + 6)i (3x + 2i + 10)i i=0

(2 n/2 ā’ 1)!!

(x + 2i)! (x + 2i + 4)!

i=0

(x2 + (4n + 3)x + 2(n2 + 4n + 1)) n even

1

Ć— Ā· (3.45)

(x + 1)(x + 2) (2x + n + 4)(2x + 2n + 4) n odd.

One of the most embarrassing failures of āidentiļ¬cation of factors,ā respectively of

LU-factorization, is the problem of q-enumeration of totally symmetric plane partitions,

as stated for example in [164, p. 289] or [165, p. 106]. It is now known for quite a

while that also this problem can be reduced to the evaluation of a certain determinant,

by means of Okadaā™s result [123, Theorem 4] about the sum of all minors of a given

matrix, that was already mentioned in Section 2.8. In fact, in [123, Theorem 5], Okada

succeeded to transform the resulting determinant into a reasonably simple one, so that

the problem of q-enumerating totally symmetric plane partitions reduces to resolving

the following conjecture.

Conjecture 46. For any nonnegative integer n there holds

2

1 ā’ q i+j+kā’1

(1) (2)

det Tn + Tn = , (3.46)

1 ā’ q i+j+kā’2

1ā¤i,jā¤n

1ā¤iā¤jā¤kā¤n

ADVANCED DETERMINANT CALCULUS 43

where

i+jā’2 i+jā’1

(1)

q i+jā’1

Tn = +q

iā’1 i

q q 1ā¤i,jā¤n

and ļ£« ļ£¶

1+q

ļ£¬ ā’1 1 + q 2 ļ£·

0

ļ£¬ ļ£·

ļ£¬ ļ£·

ā’1 1 + q 3

Tn = ļ£¬ ļ£·.

(2)

ļ£¬ ā’1 1 + q ļ£·

4

ļ£¬ ļ£·

ļ£ ļ£ø

.. ..

. .

0

ā’1 1 + q n

While the problem of (plain) enumeration of totally symmetric plane partitions was

solved a few years ago by Stembridge [170] (by some ingenious transformations of the

determinant which results directly from Okadaā™s result on the sum of all minors of a

matrix), the problem of q-enumeration is still wide open. āIdentiļ¬cation of factorsā

cannot even get started because so far nobody came up with a way of introducing a

parameter in (3.46) or any equivalent determinant (as it turns out, the parameter q

cannot serve as a parameter in the sense of Section 2.4), and, apparently, guessing the

LU-factorization is too diļ¬cult.

Let us proceed by giving a few more determinants which arise in the enumeration of

rhombus tilings.

Our next determinant evaluation is the evaluation of a determinant which, on dis-

regarding the second binomial coeļ¬cient, would be just a special case of (3.13), and

which, on the other hand, resembles very much the q = 1 case of (3.18). (It is the

determinant that was shown as (1.4) in the Introduction.) However, neither Lemma 3

nor Lemma 5 suļ¬ce to give a proof. The proof in [30] by means of āidentiļ¬cation of

factorsā is unexpectedly diļ¬cult.

Theorem 47. Let n be a positive integer, and let x and y be nonnegative integers.

Then the following determinant evaluation holds:

x+y+j x+y+j

ā’

det

x ā’ i + 2j x + i + 2j

1ā¤i,jā¤n

n

(j ā’ 1)! (x + y + 2j)! (x ā’ y + 2j + 1)j (x + 2y + 3j + 1)nā’j

= . (3.47)

(x + n + 2j)! (y + n ā’ j)!

j=1

This determinant evaluation is used in [30] to enumerate rhombus tilings of a certain

pentagonal subregion of a hexagon.

To see an example of diļ¬erent nature, I present a determinant evaluation from [50,

Lemma 2.2], which can be considered as a determinant of a mixture of two matrices,

out of one we take all rows except the l-th, while out of the other we take just the l-th

row. The determinants of both of these matrices could be straightforwardly evaluated

by means of Lemma 3. (They are in fact equivalent to special cases of (3.13).) However,

to evaluate this mixture is much more challenging. In particular, the mixture does not

44 C. KRATTENTHALER

anymore factor completely into āniceā factors, i.e., the result is of the form (2.21), so

that for a proof one has to resort to the extension of āidentiļ¬cation of factorsā described

there.

Theorem 48. Let n, m, l be positive integers such that 1 ā¤ l ā¤ n. Then there holds

ļ£«ļ£± ļ£¶

ļ£² n+mā’i (m+ 2 ) if i = l

nā’j+1

det ļ£ m+iā’j (n+jā’2i+1) ļ£ø

ļ£³ n+mā’i

1ā¤i,jā¤n

if i = l

m+iā’j

n/2

n

(n + m ā’ i)! 1

= (m + i)nā’2i+1 (m + i + )nā’2i

(m + i ā’ 1)! (2n ā’ 2i + 1)! 2

i=1 i=1

ā’ 1)!

n lā’1

(n ā’ 2e) ( 1 )e

(m)n+1 j=1 (2j n

(nā’1)(nā’2)

Ć—2 e 2

(ā’1) . (3.48)

2

e (m + e) (m + n ā’ e) ( 1 ā’ n)e

n/2

n! (2i)2nā’4i+1 2

e=0

i=1

In [50], this result was applied to enumerate all rhombus tilings of a symmetric

hexagon that contain a ļ¬xed rhombus. In Section 4 of [50] there can be found several

conjectures about the enumeration of rhombus tilings with more than just one ļ¬xed

rhombus, all of which amount to evaluating other mixtures of the above-mentioned two

determinants.

As last binomial determinants, I cannot resist to show the, so far, weirdest deter-

minant evaluations that I am aware of. They arose in an attempt [16] by Bombieri,

Hunt and van der Poorten to improve on Thueā™s method of approximating an algebraic

number. In their paper, they conjectured the following determinant evaluation, which,

thanks to van der Poorten [132], has recently become a theorem (see the subsequent

paragraphs and, in particular, Theorem 51 and the remark following it).

Theorem 49. Let N and l be positive integers. Let M be the matrix with rows labelled

by pairs (i1, i2) with 0 ā¤ i1 ā¤ 2l(N ā’ i2) ā’ 1 (the intuition is that the points (i1, i2 ) are

the lattice points in a right-angled triangle), with columns labelled by pairs (j1, j2 ) with

0 ā¤ j2 ā¤ N and 2l(N ā’ j2 ) ā¤ j1 ā¤ l(3N ā’ 2j2 ) ā’ 1 (the intuition is that the points

(j1, j2 ) are the lattice points in a lozenge), and entry in row (i1, i2) and column (j1, j2 )

equal to

j1 j2

.

i1 i2

Then the determinant of M is given by

(N3 )

+2

lā’1 3lā’1

k=0 k! k=2l k!

Ā± .

2lā’1

k=l k!

This determinant evaluation is just one in a whole series of conjectured determinant

evaluations and greatest common divisors of minors of a certain matrix, many of them

reported in [16]. These conjectures being settled, the authors of [16] expect important

implications in the approximation of algebraic numbers.

The case N = 1 of Theorem 49 is a special case of (3.11), and, thus, on a shallow

level. On the other hand, the next case, N = 2, is already on a considerably deeper

ADVANCED DETERMINANT CALCULUS 45

level. It was ļ¬rst proved in [94], by establishing, in fact, a much more general result,

given in the next theorem. It reduces to the N = 2 case of Theorem 49 for x = 0,

b = 4l, and c = 2l. (In fact, the x = 0 case of Theorem 50 had already been conjectured

in [16].)

Theorem 50. Let b, c be nonnegative integers, c ā¤ b, and let ā(x; b, c) be the determi-

nant of the (b + c) Ć— (b + c) matrix

ļ£« 0ā¤j<c c ā¤ j <b b ā¤ j <b+cļ£¶

. .

. .

. . 2x + j

ļ£¬ ļ£·

. .

x+j x+j

. .

0ā¤i<c ļ£¬ ļ£·

. .

. .

ļ£¬............................................................ ļ£·

. .

. .

i i i

. .

ļ£¬ ļ£·

. .

. .

ļ£¬ ļ£·

. . 2x + j

. .

ļ£¬ ļ£·.

x+j

. . (3.49)

cā¤i<b 0 . .

ļ£¬ ļ£·

. .

. .

ļ£¬............................................................ ļ£·

. .

i i

. .

ļ£¬ ļ£·

. .

. .

ļ£ ļ£ø

. .

. .

x+j . x+j .

bā¤i <b+c 0

. .

2 . .

iā’b . .

iā’b

. .

. .

Then

(i) ā(x; b, c) = 0 if b is even and c is odd;

(ii) if any of these conditions does not hold, then

i+ 1 ā’

bā’c b

2 2

cc c

ā(x; b, c) = (ā’1) 2

(i)c

i=1

+ c+i bā’c+i

c x x+

bā’c+ i/2 ā’ (c+i)/2 (b+i)/2 ā’ (bā’c+i)/2

2 2

Ć— .

ā’ ā’

1 b

+ c+i 1 b bā’c+i

+

bā’c+ i/2 ā’ (c+i)/2 (b+i)/2 ā’ (bā’c+i)/2

2 2 2 2 2 2

i=1

(3.50)

The proof of this result in [94] could be called āheavyā. It proceeded by āidentiļ¬cation

of factorsā. Thus, it was only the introduction of the parameter x in the determinant in

(3.49) that allowed the attack on this special case of the conjecture of Bombieri, Hunt

and van der Poorten. However, the authors of [94] (this includes myself) failed to ļ¬nd a

way to introduce a parameter into the determinant in Theorem 49 for generic N (that

is, in a way such the determinant would still factor nicely). This was accomplished by

van der Poorten [132]. He ļ¬rst changed the entries in the determinant slightly, without

changing the value of the determinant, and then was able to introduce a parameter. I

state his result, [132, Sec. 5, Main Theorem], in the theorem below. For the proof of

his result he used āidentiļ¬cation of factorsā as well, thereby considerably simplifying

and illuminating arguments from [94].

Theorem 51. Let N and l be positive integers. Let M be the matrix with rows labelled

by pairs (i1 , i2) with 0 ā¤ i1 ā¤ 2l(N ā’ i2 ) ā’ 1, with columns labelled by pairs (j1 , j2) with

0 ā¤ j2 ā¤ N and 0 ā¤ j1 ā¤ lN ā’ 1, and entry in row (i1, i2) and column (j1 , j2 ) equal to

ā’x(N ā’ j2 ) j2

(ā’1)i1 ā’j1 . (3.51)

i1 ā’ j1 i2

46 C. KRATTENTHALER

Then the determinant of M is given by

(N3 )

+2

l

x+iā’1 l+iā’1

Ā± .

2i ā’ 1 2i ā’ 1

i=1

Although not completely obvious, the special case x = ā’2l establishes Theorem 49,

see [132]. Van der Poorten proves as well an evaluation that overlaps with the x = 0

case of Theorem 50, see [132, Sec. 6, Example Application].

Let us now turn to a few remarkable Hankel determinant evaluations.

Theorem 52. Let n be a positive integer. Then there hold

nā’1

(2i)!2,

det (E2i+2j ) = (3.52)

0ā¤i,jā¤nā’1

i=0

where E2k is the (2k)-th (signless) Euler number, deļ¬ned through the generating function

1/ cos z = ā E2k z 2k /(2k)!, and

k=0

nā’1

(2i + 1)!2 .

det (E2i+2j+2 ) = (3.53)

0ā¤i,jā¤nā’1

i=0

Furthermore, deļ¬ne the Bell polynomials Bm (x) by Bm (x) = m S(m, k)xk , where

k=1

S(m, k) is a Stirling number of the second kind (the number of partitions of an m-

element set into k blocks; cf. [166, p. 33]). Then

nā’1

n(nā’1)/2

det (Bi+j (x)) = x i!. (3.54)

0ā¤i,jā¤nā’1

i=0

Next, there holds

nā’1

n(nā’1)/2

det (Hi+j (x)) = (ā’1) i!, (3.55)

0ā¤i,jā¤nā’1

i=0

k

where Hm (x) = kā„0 k! (mā’2k)! ā’ 1 xmā’2k is the m-th Hermite polynomial.

m!

2

Finally, the following Hankel determinant evaluations featuring Bernoulli numbers

hold,

nā’1

i!6

n

(Bi+j ) = (ā’1)( )

det , (3.56)

2

(2i)! (2i + 1)!

0ā¤i,j,ā¤nā’1

i=1

and

nā’1

i!3 (i + 1)!3

)1

n+1

(Bi+j+1 ) = (ā’1)(

det , (3.57)

2

2 (2i + 1)! (2i + 2)!

0ā¤i,j,ā¤nā’1

i=1

and

nā’1

i! (i + 1)!4 (i + 2)!

1 n

det (Bi+j+2 ) = (ā’1)( ) , (3.58)

2

6 (2i + 2)! (2i + 3)!

0ā¤i,j,ā¤nā’1

i=1

ADVANCED DETERMINANT CALCULUS 47

and

nā’1

(2i)! (2i + 1)!4 (2i + 2)!

det (B2i+2j+2 ) = , (3.59)

(4i + 2)! (4i + 3)!

0ā¤i,j,ā¤nā’1

i=0

and

n

(2i ā’ 1)! (2i)!4 (2i + 1)!

n

det (B2i+2j+4 ) = (ā’1) . (3.60)

(4i)! (4i + 1)!

0ā¤i,j,ā¤nā’1

i=1

All these evaluations can be deduced from continued fractions and orthogonal poly-

nomials, in the way that was described in Section 2.7. To prove (3.52) and (3.53)

one would resort to suitable special cases of Meixnerā“Pollaczek polynomials (see [81,

Sec. 1.7]), and use an integral representation for Euler numbers, given for example in

ńņš. 2 |