<<

. 2
( 3)



>>

(mi )
(Xj ’ Xi )mi mj .
2
det Bm1 (X1 ) Bm2 (X2 ) . . . Bm (X ) = Xi j!
1¤i,j,¤n
i=1 j=1 1¤i<j¤
(3.2)


As Alain Lascoux taught me, the natural environment for this type of determinants
is divided di¬erences and (generalized) discrete Wronskians. The divided di¬erence ‚x,y
is a linear operator which maps polynomials in x and y to polynomials symmetric in x
and y, and is de¬ned by
f(x, y) ’ f(y, x)
‚x,y f(x, y) = .
x’y
Divided di¬erences have been introduced by Newton to solve the interpolation prob-
lem in one variable. (See [100] for an excellent introduction to interpolation, divided
di¬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 coe¬cients do not depend on a1, a2, . . . , am ,
Newton™s interpolation formula reads as follows (cf. e.g. [100, (Ni2)]),
g(x) = g(a1 ) + (x ’ a1)‚a1 ,a2 g(a1) + (x ’ a1)(x ’ a2 )‚a2 ,a3 ‚a1 ,a2 g(a1 )
+ (x ’ a1 )(x ’ a2 )(x ’ a3 )‚a3,a4 ‚a2 ,a3 ‚a1 ,a2 g(a1) + · · · . (3.3)
Now suppose that f1 (x), f2(x), . . . , fn (x) are polynomials in one variable x, whose
coe¬cients do not depend on a1, a2, . . . , an , and consider the determinant
det (fi (aj )). (3.4)
1¤i,j,¤n

Let us for the moment concentrate on the ¬rst m1 columns of this determinant. We
may apply (3.3), and write
fi (aj ) = fi (a1) + (aj ’ a1)‚a1 ,a2 fi (a1) + (aj ’ a1)(aj ’ a2)‚a2 ,a3 ‚a1 ,a2 fi (a1)
+ · · · + (aj ’ a1)(aj ’ a2) · · · (aj ’ aj’1 )‚aj’1 ,aj · · · ‚a2 ,a3 ‚a1,a2 fi (a1 ),
j = 1, 2, . . . , m1. Following [100, Proof of Lemma (Ni5)], we may perform column
reductions to the e¬ect that the determinant (3.4), with column j replaced by
(aj ’ a1 )(aj ’ a2 ) · · · (aj ’ aj’1 )‚aj’1 ,aj · · · ‚a2,a3 ‚a1 ,a2 fi (a1),
j = 1, 2, . . . , m1, has the same value as the original determinant. Clearly, the product
k=1 (aj ’ ak ) can be taken out of column j, j = 1, 2, . . . , m1 . Similar reductions can
j’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 — m matrix ‚xj’1 ,xj · · · ‚x2 ,x3 ‚x1 ,x2 fi (x1) 1¤i¤n, 1¤j¤m . Given a composition of n,
n = m1 + · · · + m , there holds
det Wm1 (a1, . . . , am1 ) Wm2 (am1 +1 , . . . , am1 +m2 ) . . . Wm (am1 +···+m ’1 +1 , . . . , an )
1¤i,j,¤n


(aj ’ ai ) . (3.5)
= det (fi (aj ))
1¤i,j,¤n
m1 +···+mk’1 +1¤i<j¤m1 +···+mk
k=1



If we now choose fi (x) := xi’1 , so that det1¤i,j,¤n (fi (aj )) is a Vandermonde deter-
minant, then the right-hand side of (3.5) factors completely by (2.1). The ¬nal step
to obtain Theorem 20 is to let a1 ’ X1 , a2 ’ X1 , . . . , am1 ’ X1 , am1 +1 ’ X2 , . . . ,
am1 +m2 ’ X2 , etc., in (3.5). This does indeed yield (3.1), because
j’1
1 d
lim . . . lim lim ‚xj’1 ,xj · · · ‚x2 ,x3 ‚x1 ,x2 g(x1) = g(x),
(j ’ 1)! dx
xj ’x x2 ’x x1 ’x

as is easily veri¬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 ’1 for j = m1 +1, m1 +2, . . . , m2 ,
j’1 j’m

etc., and by then using the relation
d d
Xg(X) ’ g(X)
X g(X) =
dX dX
28 C. KRATTENTHALER

j’1 i’1
many times, so that a typical entry Xk (d/dXk )j’1 Xk in row i and column j of the
i’1
k-th submatrix is expressed as (Xk (d/dXk ))j’1 Xk plus a linear combination of terms
(Xk (d/dXk ))s Xk with s < j ’ 1. Simple column reductions then yield (3.2).
i’1


It is now not very di¬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 — m matrix
«
[C]q X ’1 [C]q [C ’ 1]q X ’2
1
¬X [C + 1]q [C]q X ’1
[C + 1]q
¬2
¬X [C + 2]q X [C + 2]q [C + 1]q
¬
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
X n’1 [C + n ’ 1]q X n’2 [C + n ’ 1]q [C + n ’ 2]q X n’3

[C]q · · · [C ’ m + 2]q X 1’m
...
·
[C + 1]q · · · [C ’ m + 3]q X 2’m
... ·
·,
[C + 2]q · · · [C ’ m + 4]q X 3’m
... ·
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . [C + n ’ 1]q · · · [C + n ’ m + 1]q X n’m

i.e., any next column is formed by applying the operator X ’C Dq X C , with Dq denoting
the usual q-derivative, Dq f(X) := (f(qX) ’ f(X))/(q ’ 1)X. Given a composition of
n, n = m1 + · · · + m , there holds

det Am1 (X1 ) Am2 (X2 ) . . . Am (X )
1¤i,j,¤n
mi ’1 mj ’1
mi ’1
(q t’s Xj ’ Xi ), (3.6)
= q N1 [j]q !
i=1 j=1 s=0 t=0
1¤i<j¤

where N1 is the quantity
mi
(C + j + m1 + · · · + mi’1 ’ 1)(mi ’ j) ’ ’ ’ mj
mi mj mi
mi .
3 2 2
i=1 j=1 1¤i<j¤




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

repeatedly.

A “q-Abel-type” 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 — m matrix
« 
[C]2 [C]m’1
1 [C]q ...
q q
¬X ·
2
[C + 1]m’1 X
[C + 1]q X [C + 1]q X ...
¬2 ·
q
¬X [C + 2]m’1 X 2 · ,
[C + 2]q X 2 [C + 2]2 X 2 ...
¬ ·
q q
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
X n’1 [C + n ’ 1]q X n’1 [C + n ’ 1]2 X n’1 . . . [C + n ’ 1]m’1 X n’1 q q

i.e., any next column is formed by applying the operator X 1’C Dq X C , with Dq denoting
the q-derivative as in Theorem 23. Given a composition of n, n = m1 + · · · + m , there
holds

det Bm1 (X1 ) Bm2 (X2 ) . . . Bm (X )
1¤i,j,¤n
mi ’1 mj ’1
mi ’1
(mi )
(q t’s Xj ’ Xi ), (3.8)
= q N2 2
Xi [j]q !
i=1 j=1 s=0 t=0
1¤i<j¤

where N2 is the quantity
mi
((C + j + m1 + · · · + mi’1 ’ 1)(mi ’ j)) ’ ’ mj
mj mi
mi .
2 2
i=1 j=1 1¤i<j¤




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 Cauchy™s double alternant (2.7) can also be found in the literature (see
e.g. [117, 149]). I want to mention here particularly Borchardt™s variation [17] in which
the (i, j)-entry in Cauchy™s double alternant is replaced by its square,
’ Xj )(Yi ’ Yj )
1¤i<j¤n (Xi
1 1
det = Per , (3.9)
(Xi ’ Yj )2 1¤i,j¤n (Xi ’ Yj )
1¤i,j¤n Xi ’ Yj
1¤i,j¤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 “closed form evalu-
ation,” thus solving a famous enumeration problem in an unexpected way, the problem
of enumerating alternating sign matrices.10 This q-deformation is equivalent to Izergin™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, ’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 (’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 “birth” of alternating sign matrices came through ” determinants, see
[150].
30 C. KRATTENTHALER

Theorem 25. For any nonnegative integer n there holds
’ Xj )(Yi ’ Yj )
1¤i<j¤n (Xi
1
det =
(Xi ’ Yj )(qXi ’ Yj ) 1¤i,j¤n (Xi ’ Yj )(qXi ’ Yj )
1¤i,j¤n
n
N i (A)
Ni (A)
— (1 ’ q) (±i,j Xi ’ Yj ), (3.10)
2N (A)
Xi Yi
i=1
A i,j such that Aij =0

where the sum is over all n — n alternating sign matrices A = (Aij )1¤i,j¤n , N(A) is
the number of (’1)s in A, Ni (A) (respectively N i (A)) is the number of (’1)s in the
i-th row (respectively column) of A, and ±ij = q if j Aik = i Akj , and ±ij = 1
k=1 k=1
otherwise.
Clearly, equation (3.9) results immediately from (3.10) by setting q = 1. Roughly,
Kuperberg™s solution [97] of the enumeration of alternating sign matrices consisted of
suitably specializing the xi ™s, yi ™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 — n alternating sign matrices, and in evaluating the left-hand side determinant for
that special choice of the xi ™s, yi ™s and q. The resulting number of n — n alternating
sign matrices is given in (A.1) in the Appendix. (The ¬rst, very di¬erent, solution
is due to Zeilberger [198].) Subsequently, Zeilberger [199] improved on Kuperberg™s
approach and succeeded in proving the re¬ned alternating sign matrix conjecture from
[111, Conj. 2]. For a di¬erent expansion of the determinant of Izergin, in terms of Schur
functions, and a variation, see [101, Theorem q, Theorem γ].
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
1¤i<j¤n [Li ’ Lj ]q
n
i=1 [Li + A + 1]q !
Li + A + j n
(i’1)(Li +i)
det = q i=1 ,
[A + 1 ’ i]q !
n n
Li + j [Li + n]q !
1¤i,j¤n
q i=1 i=1
(3.11)
and
1¤i<j¤n [Li ’ Lj ]q i=1 [A + i ’ 1]q !
n
A n
jLi iLi
det q =q , (3.12)
i=1
[A ’ Li ’ 1]q !
n n
Li + j i=1 [Li + n]q !
1¤i,j¤n
q i=1

and
BLi + A
det
Li + j
1¤i,j¤n

1¤i<j¤n (Li ’ Lj )
n n
(BLi + A)!
(A ’ Bi + 1)i’1 , (3.13)
=
((B ’ 1)Li + A ’ 1)!
n
i=1 (Li + n)! i=1 i=1
and
n
(A + BLi )j’1 (A + Bi)i’1
(Lj ’ Li ).
det = (3.14)
(j ’ Li )! (n ’ Li )!
1¤i,j¤n
i=1 1¤i<j¤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
’A ’ 1
Li + A + j Li j
= (’1)Li +j q ( 2 )+(2)+jLi +(A+1)(Li +j) .
Li + j Li + j q
q

Hence, replacement of A by ’A ’ 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 ¬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 Jacobi“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)“(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]).
MacMahon™s “box formula” [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 »-
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 “semiregular” 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 ’ j
q jLi
det
Li + j
1¤i,j¤n
q
n
[Li + A ’ n]q !
n
[Li ’ Lj ]q [Li + Lj + A + 1]q . (3.15)
iLi
=q i=1
[Li + n]q ! [A ’ 2i]q ! 1¤i<j¤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 28“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 di¬erence being that Lemma 5 is invoked instead of Lemma 3).
The ¬rst application is the evaluation of a determinant whose entries are a product
of two q-binomial coe¬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 ’ j
Li + j
·
det
B B
1¤i,j¤n
q q

( )+2(n+1)
n
i=1 (i’1)Li ’B 2
n
[Li ’ Lj ]q [Li + Lj + A ’ B + 1]q
=q 3

1¤i<j¤n
n
[Li + 1]q ! [Li + A ’ n]q ! [A ’ 2i ’ 1]q !
— . (3.16)
[Li ’ B + n]q ! [Li + A ’ B ’ 1]q ! [A ’ i ’ n ’ 1]q ! [B + i ’ n]q ! [B]q !
i=1




As is not di¬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 ¬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 , . . . , Xn’1 , Y0 , Y1 , . . . ,
Yn’1 , A and B be indeterminates. Then there holds
« 
Yj + A ’ Xi
Xi + Yj
¬ ·
j j
¬ q·
q
det ¬ ·
0¤i,j¤n’1  Xi + B A + B ’ Xi 
j j
q q
n n’1
i(Xi +Yi ’A’2B)
= q 2(3 )+ [Xi ’ Xj ]q [Xi + Xj ’ A]q
i=0

0¤i<j¤n’1
n’1
(q B’Yi ’i+1 )i (q Yi +A+B+2’2i )i
— . (3.17)
(q Xi’A’B )n’1 (q Xi+B’n+2 )n’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 di¬erence of two q-binomial coe¬cients.
A proof of the ¬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, §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 i’1/2, i = 1, 2, . . . ) of a symplectic character of arbitrary shape, by comparing the
symplectic dual Jacobi“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 ’Li ) ’ q j(2Li+A’1)
det
j ’ Li ’j ’ Li + 1
1¤i,j¤n
q q
n
[A + 2i ’ 2]q !
[Lj ’ Li ]q [Li + Lj + A ’ 1]q (3.18)
=
[n ’ Li ]q ! [A + n ’ 1 + Li ]q ! 1¤i<j¤n
i=1 1¤i¤j¤n

and
A A
q j(Lj ’Li ) ’ q j(2Li+A)
det
j ’ Li ’j ’ Li
1¤i,j¤n
q q
n
[A + 2i ’ 1]q !
[Lj ’ Li ]q
= [Li + Lj + A]q . (3.19)
[n ’ Li ]q ! [A + n + Li ]q ! 1¤i<j¤n
i=1 1¤i¤j¤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, Andrews™ evaluation
proceeded by LU-factorization, while Schlosser [155, Ch. VI, §3] simpli¬ed Andrews™
proof signi¬cantly by making use of Lemma 5. The determinant evaluation (3.18)
34 C. KRATTENTHALER

was used in [88] in the proof of re¬nements of the MacMahon (ex-)Conjecture and the
Bender“Knuth (ex-)Conjecture. (The latter makes an assertion about the generating
function for tableaux with bounded entries and a bounded number of columns. The
¬rst proof is due to Gordon [59], the ¬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 coe¬cients. A proof of the ¬rst evaluation which uses Lemma 5 can be found
in [155, Ch. VI, §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
Jacobi“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 (j’1/2)(Lj ’Li ) + q (j’1/2)(2Li+A’1)
det
j ’ Li ’j ’ Li + 1
1¤i,j¤n
q q
n
[A + 2i ’ 1]q !
(1 + q Li +A/2’1/2)
=
(1 + q i+A/2’1/2) [n ’ Li ]q ! [A + n + Li ’ 1]q !
i=1

— [Lj ’ Li ]q [Li + Lj + A ’ 1]q (3.20)
1¤i<j¤n

and
A A
q (j’1/2)(Lj ’Li ) + q (j’1/2)(2Li+A’2)
det
j ’ Li ’j ’ Li + 2
1¤i,j¤n
q q
n
[A + 2i ’ 2]q !
n Li+A/2’1
i=1 (1 + q )
=
[n ’ Li ]q ! [A + n + Li ’ 2]q !
n i+A/2’1 )
i=2 (1 + q i=1

— [Lj ’ Li ]q [Li + Lj + A ’ 2]q . (3.21)
1¤i<j¤n




A special case of (3.20) was the ¬rst determinant evaluation which Andrews needed
in [4, (1.3)] in order to prove the MacMahon Conjecture on symmetric plane parti-
tions. Again, Andrews™ evaluation proceeded by LU-factorization, while Schlosser [155,
Ch. VI, §3] simpli¬ed Andrews™ proof signi¬cantly by making use of Lemma 5.
Now we come to determinants which belong to a di¬erent category what regards
di¬culty of evaluation, as it is not possible to introduce more parameters in a substantial
way.
The ¬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, Macdonald™s problem of
ADVANCED DETERMINANT CALCULUS 35

enumerating cyclically symmetric plane partitions and Andrews™ own conjecture about
the enumeration of descending plane partitions.
Theorem 32. Let µ be an indeterminate. For nonnegative integers n there holds
2µ + i + j
det δij +
j
0¤i,j¤n’1
± n’2
 n/2
2

 (µ + i/2 + 1) (i+3)/4



 i=1


i=1 µ + 2 ’ 2 + 2 i/2 ’1 µ + 2 ’ 2 +
n/2
 3n 3i 3 3n 3i 3

— 2 i/2
 if n is even,

i=1 (2i ’ 1)!! (2i + 1)!!
n/2’1
=
 n/2 n’2

2
 (µ + i/2 + 1) (i+3)/4




 i=1

 µ + 3n ’ 3i’1 + 1 (i’1)/2 µ + 3n ’
(n’1)/2 3i

— i=1
 2 2 2 2 i/2
 if n is odd.
(2i ’ 1)!!2
(n’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 µ = 0 and µ = 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 ’ q 3i’1 1 ’ q 3(n+i+j’1)
i+j
3i+1
det δij + q = . (3.23)
1 ’ q 3i’2 1¤i¤j¤n 1 ’ q 3(2i+j’1)
j
0¤i,j¤n’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 ’ q n+i+j
i+j +2
i+2
det δij + q = . (3.24)
1 ’ q 2i+j’1
j
0¤i,j¤n’1
q 1¤i¤j¤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 µ = 0 and µ = 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 coe¬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 µ be an indeterminate. For nonnegative integers n there holds
2µ + i + j
’δij +
det
j
0¤i,j¤n’1

0, if n is odd,
= (3.25)
i!2 (µ+i)!2 (µ+3i+1)!2 (2µ+3i+1)!2
n/2’1
(’1)n/2 , if n is even.
(2i)! (2i+1)! (µ+2i)!2 (µ+2i+1)!2 (2µ+2i)! (2µ+2i+1)!
i=0

If ω is a primitive 3rd root of unity, then for nonnegative integers n there holds
(1 + ω)n2 n/2
2µ + i + j
det ωδij + =
(2i ’ 1)!! (2i ’ 1)!!
n/2 (n’1)/2
j
0¤i,j¤n’1
i=1 i=1

— (µ + 3i + 1) (µ + 3i + 3)
(n’4i)/2 (n’4i’3)/2
i≥0

· µ+n’i+ µ+n’i’
1 1
, (3.26)
2 2
(n’4i’1)/2 (n’4i’2)/2

where, in abuse of notation, by ± we mean the usual ¬‚oor function if ± ≥ 0, however,
if ± < 0 then ± must be read as 0, so that the product over i in (3.26) is indeed a
¬nite product.
If ω is a primitive 6th root of unity, then for nonnegative integers n there holds
2 n/2
(1 + ω)n
2µ + i + j 3
det ωδij + =
(2i ’ 1)!! (2i ’ 1)!!
n/2 (n’1)/2
j
0¤i,j¤n’1
i=1 i=1

— 3 5
µ + 3i + µ + 3i +
2 2
(n’4i’1)/2 (n’4i’2)/2
i≥0

· (µ + n ’ i) (µ + n ’ i) , (3.27)
(n’4i)/2 (n’4i’3)/2

where again, in abuse of notation, by ± we mean the usual ¬‚oor function if ± ≥ 0,
however, if ± < 0 then ± must be read as 0, so that the product over i in (3.27) is
indeed a ¬nite product.
There are no really simple proofs of Theorems 32“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 “easiest” way of evaluating
the determinant (3.22) is by ¬rst employing a magni¬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; µ, ν) be de¬ned by
n’1 n’1
j ’ k + µ ’ 1 k’t
i+µ k+ν
Zn (x; µ, ν) := det δij + x ,
k’t j’k
t
0¤i,j¤n’1
t=0 k=0

let Tn (x; µ, ν) be de¬ned by
2j
i+µ j+ν
x2j’t ,
Tn (x; µ, ν) := det
t’i 2j ’ t
0¤i,j¤n’1
t=i

and let Rn (x; µ, ν) be de¬ned by
2j+1
i+µ i+µ+1
Rn (x; µ, ν) := det +
t’i’1 t’i
0¤i,j¤n’1
t=i

j+ν j+ν+1
· x2j+1’t .
+
2j + 1 ’ t 2j + 1 ’ t

Then for all positive integers n there hold
Z2n(x; µ, ν) = Tn (x; µ, ν/2) Rn (x; µ, ν/2) (3.28)
and
Z2n’1 (x; µ, ν) = 2 Tn(x; µ, ν/2) Rn’1 (x; µ, ν/2). (3.29)


The reader should observe that Zn (1; µ, 0) is identical with the determinant in (3.22),
as the sums in the entries simplify by means of Chu“Vandermonde summation (see e.g.
[62, Sec. 5.1, (5.27)]). However, also the entries in the determinants Tn (1; µ, 0) and
Rn(1; µ, 0) simplify. The respective evaluations read as follows (see [112, Theorem 7]
and [9, (5.2)/(5.3)]).
Theorem 37. Let µ be an indeterminate. For nonnegative integers n there holds
µ+i+j
det
2i ’ j
0¤i,j¤n’1

’µ ’ 3n + i + 3
n’1 (µ + i + 1) (i+1)/2
n’1 2
2( ) i/2
= (’1)χ(n≡3 mod 4)
, (3.30)
2
(i)i
i=1

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

µ+i+j µ+i+j +2
det +2
2i ’ j 2i ’ j + 1
0¤i,j¤n’1
n
(µ + 3n ’ 3i’1 + 1 )
(µ + i) i/2 (i+1)/2
n 2 2
=2 . (3.31)
(2i ’ 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 Andrews™ Theorem 32. However, in the meantime there exist short
direct proofs of (3.30), see [10, 91, 129], either by LU-factorization, or by “identi¬cation
of factors”. 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; µ, ν)
and Tn (x; µ, ν) 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; µ, ν), Tn (x; µ, ν), Rn (x; µ, ν) do
also factor nicely for x = 2. This was proved by Andrews [7] using LU-factorization,
thus con¬rming a conjecture by Mills, Robbins and Rumsey (see [92] for an alternative
proof by “identi¬cation of factors”).
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 ’ j 2i ’ j
0¤i,j¤n’1

’ x+y ’ 3n + i +
n’1 x+y + i + 1 3
n 2 2 2
= (’1)χ(n≡3 mod 4)2( 2 )+1
(i+1)/2 i/2
, (3.32)
(i)i
i=1

where χ(A) = 1 if A is true and χ(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 , . . . , xn’1 and c be indeterminates. For nonnegative integers
n there hold
c ’ xi + i + j
c + xi + i + j
det +
2i ’ j 2i ’ j
0¤i,j¤n’1

’c ’ 3n + i + 3
n’1 (c + i + 1)
(i+1)/2
n 2
χ(n≡3 mod 4) ( 2 )+1 i/2
= (’1) 2 (3.33)
(i)i
i=1

and
(2i ’ j) + (2c + 3j + 1)(2c + 3j ’ 1) c + i + j + 1
2
det
2i ’ j
(c + i + j + 1 )(c + i + j ’ 1 )
0¤i,j¤n’1
2 2
(’c ’ 3n + i + 2)
1
n’1 c+i+ i/2
n+1 2
2( )+1 (i+1)/2
χ(n≡3 mod 4)
= (’1) , (3.34)
2
(i)i
i=1

where χ(A) = 1 if A is true and χ(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 Mills“Robbins“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, ¬rst, that it
does not seem to be possible to introduce a second parameter into the Mills“Robbins“
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 ¬‚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 ’ 1)!
det
(x + 2i ’ j)! (y + 2j ’ i)!
0¤i,j¤n’1
n’1
i! (x + y + i ’ 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 di¬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 “identi¬cation of
factors”. However, Amdeberhan [2] observed that it can be easily proved by “conden-
sation”.
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 Mills“Robbins“
Rumsey determinant (3.30), but also simple proofs for the enumeration of four di¬erent
40 C. KRATTENTHALER

symmetry classes of plane partitions, cyclically symmetric plane partitions, cyclically
symmetric self-complementary plane partitions (¬rst proved by Kuperberg [96]), cycli-
cally symmetric transpose-complementary plane partitions (¬rst proved by Mills, Rob-
bins and Rumsey [112]), and totally symmetric self-complementary plane partitions (¬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 “condensa-
tion”.
Theorem 41. Let x, y, n be nonnegative integers. Then there holds
q ’2ij
(q; q)x+y+i+j’1
det
(q; q)x+2i’j (q; q)y+2j’i (’q x+y+1 ; q)i+j
0¤i,j¤n’1
n’1 2
; q 2)i (q; q)x+y+i’1 (q 2x+y+2i ; q)i (q x+2y+2i ; q)i
’2i2 (q
= q . (3.36)
(q; q)x+2i (q; q)y+2i (’q x+y+1 ; q)n’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 (’q x+y+1 ; q)i+j in the determinant, respectively (’q x+y+1 ; q)n’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 “´tude” e
on the Andrews and the Mills“Robbins“Rumsey determinant.
Theorem 42. Let x and E be indeterminates and n be a nonnegative integer. Then
there holds
(E/xq i ; q 2)i’j (q/Exq i; q 2)i’j (1/x2 q 2+4i; q 2)i’j
det
(q; q)2i+1’j (1/Exq 2i; q)i’j (E/xq 1+2i; q)i’j
0¤i,j¤n’1
n’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 ’ 1)! (y ’ x + 3j ’ 3i)
det
(x + 2i ’ j + 1)! (y + 2j ’ i + 1)!
0¤i,j¤n’1
n’1
i! (x + y + i ’ 1)! (2x + y + 2i + 1)i (x + 2y + 2i + 1)i
=
(x + 2i + 1)! (y + 2i + 1)!
i=0
n
n
· (’1)k (x)k (y)n’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+j’1 (1 ’ q y+2j’i ’ q y+2j’i+1 + q x+y+i+j+1 )
det
(q; q)x+2i’j+1 (q; q)y+2j’i+1
0¤i,j¤n’1

q ’2ij
·
(’q x+y+2 ; q)i+j
n’1 2
; q 2)i (q; q)x+y+i’1 (q 2x+y+2i+1 ; q)i (q x+2y+2i+1 ; q)i
’2i2 (q
= q
(q; q)x+2i+1 (q; q)y+2i+1 (’q x+y+2 ; q)n’1+i
i=0
n
n
— (’1)k q nk q yk (q x; q)k (q y ; q)n’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 “condensation”. 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].
¤ n. Under the convention
Theorem 45. Let x, m, n be nonnegative integers with m
that sums are interpreted by
±
 B
 r=A+1 Expr(r) A<B
B
Expr(r) = 0 A=B

’ A
r=A+1
r=B+1 Expr(r) A > B,
there holds
2x + m + i + j
det
r
0¤i,j¤n’1
x+2i’j<r¤x+m+2j’i
n’1
(2x + m + i)! (3x + m + 2i + 2)i (3x + 2m + 2i + 2)i
=
(x + 2i)! (x + m + 2i)!
i=1
n/2 ’1
(2x + m)!
— · (2x + 2 m/2 + 2i + 1) · 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 ¤ m/2 .
In particular, for m = 0 the determinant equals
±
 n/2’1
n’1
 (2x + 2i + 1)
 i! (2x + i)! (3x + 2i + 2)2 i=0
i
n even (3.41)
(n ’ 1)!!
 i=0 (x + 2i)!2



0 n odd,
42 C. KRATTENTHALER

for m = 1, n ≥ 1, it equals
n/2 ’1
(2x + 2i + 3)
n’1
i! (2x + i + 1)! (3x + 2i + 3)i (3x + 2i + 4)i i=0
, (3.42)
(2 n/2 ’ 1)!!
(x + 2i)! (x + 2i + 1)!
i=0
for m = 2, n ≥ 2, it equals
n/2 ’1
(2x + 2i + 3)
n’1
i! (2x + i + 2)! (3x + 2i + 4)i (3x + 2i + 6)i i=0
(2 n/2 ’ 1)!!
(x + 2i)! (x + 2i + 2)!
i=0

1 (x + n + 1) n even
— · (3.43)
(x + 1) (2x + n + 2) n odd,
for m = 3, n ≥ 3, it equals
n/2 ’1
(2x + 2i + 5)
n’1
i! (2x + i + 3)! (3x + 2i + 5)i (3x + 2i + 8)i i=0
(2 n/2 ’ 1)!!
(x + 2i)! (x + 2i + 3)!
i=0

1 (x + 2n + 1) n even
— · (3.44)
(x + 1) (3x + 2n + 5) n odd,
and for m = 4, n ≥ 4, it equals
n/2 ’1
(2x + 2i + 5)
n’1
i! (2x + i + 4)! (3x + 2i + 6)i (3x + 2i + 10)i i=0
(2 n/2 ’ 1)!!
(x + 2i)! (x + 2i + 4)!
i=0

(x2 + (4n + 3)x + 2(n2 + 4n + 1)) n even
1
— · (3.45)
(x + 1)(x + 2) (2x + n + 4)(2x + 2n + 4) n odd.



One of the most embarrassing failures of “identi¬cation of factors,” 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 Okada™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 ’ q i+j+k’1
(1) (2)
det Tn + Tn = , (3.46)
1 ’ q i+j+k’2
1¤i,j¤n
1¤i¤j¤k¤n
ADVANCED DETERMINANT CALCULUS 43

where
i+j’2 i+j’1
(1)
q i+j’1
Tn = +q
i’1 i
q q 1¤i,j¤n
and « 
1+q
¬ ’1 1 + q 2 ·
0
¬ ·
¬ ·
’1 1 + q 3
Tn = ¬ ·.
(2)
¬ ’1 1 + q ·
4
¬ ·
 
.. ..
. .
0
’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 Okada™s result on the sum of all minors of a
matrix), the problem of q-enumeration is still wide open. “Identi¬cation of factors”
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 di¬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 coe¬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 su¬ce to give a proof. The proof in [30] by means of “identi¬cation of
factors” is unexpectedly di¬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

det
x ’ i + 2j x + i + 2j
1¤i,j¤n
n
(j ’ 1)! (x + y + 2j)! (x ’ y + 2j + 1)j (x + 2y + 3j + 1)n’j
= . (3.47)
(x + n + 2j)! (y + n ’ 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 di¬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 “nice” factors, i.e., the result is of the form (2.21), so
that for a proof one has to resort to the extension of “identi¬cation of factors” described
there.
Theorem 48. Let n, m, l be positive integers such that 1 ¤ l ¤ n. Then there holds
«± 
 n+m’i (m+ 2 ) if i = l
n’j+1


det  m+i’j (n+j’2i+1) 
 n+m’i
1¤i,j¤n
if i = l
m+i’j
n/2
n
(n + m ’ i)! 1
= (m + i)n’2i+1 (m + i + )n’2i
(m + i ’ 1)! (2n ’ 2i + 1)! 2
i=1 i=1

’ 1)!
n l’1
(n ’ 2e) ( 1 )e
(m)n+1 j=1 (2j n
(n’1)(n’2)
—2 e 2
(’1) . (3.48)
2
e (m + e) (m + n ’ e) ( 1 ’ n)e
n/2
n! (2i)2n’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 ¬xed rhombus. In Section 4 of [50] there can be found several
conjectures about the enumeration of rhombus tilings with more than just one ¬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 Thue™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 ¤ i1 ¤ 2l(N ’ i2) ’ 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 ¤ j2 ¤ N and 2l(N ’ j2 ) ¤ j1 ¤ l(3N ’ 2j2 ) ’ 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
l’1 3l’1
k=0 k! k=2l k!
± .
2l’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 ¬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 ¤ b, and let ∆(x; b, c) be the determi-
nant of the (b + c) — (b + c) matrix

« 0¤j<c c ¤ j <b b ¤ j <b+c
. .
. .
. . 2x + j
¬ ·
. .
x+j x+j
. .
0¤i<c ¬ ·
. .
. .
¬............................................................ ·
. .
. .
i i i
. .
¬ ·
. .
. .
¬ ·
. . 2x + j
. .
¬ ·.
x+j
. . (3.49)
c¤i<b 0 . .
¬ ·
. .
. .
¬............................................................ ·
. .
i i
. .
¬ ·
. .
. .
 
. .
. .
x+j . x+j .
b¤i <b+c 0
. .
2 . .
i’b . .
i’b
. .
. .
Then
(i) ∆(x; b, c) = 0 if b is even and c is odd;
(ii) if any of these conditions does not hold, then

i+ 1 ’
b’c b
2 2
cc c
∆(x; b, c) = (’1) 2
(i)c
i=1
+ c+i b’c+i
c x x+
b’c+ i/2 ’ (c+i)/2 (b+i)/2 ’ (b’c+i)/2
2 2
— .
’ ’
1 b
+ c+i 1 b b’c+i
+
b’c+ i/2 ’ (c+i)/2 (b+i)/2 ’ (b’c+i)/2
2 2 2 2 2 2
i=1

(3.50)


The proof of this result in [94] could be called “heavy”. It proceeded by “identi¬cation
of factors”. 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 ¬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 ¬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 “identi¬cation of factors” 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 ¤ i1 ¤ 2l(N ’ i2 ) ’ 1, with columns labelled by pairs (j1 , j2) with
0 ¤ j2 ¤ N and 0 ¤ j1 ¤ lN ’ 1, and entry in row (i1, i2) and column (j1 , j2 ) equal to
’x(N ’ j2 ) j2
(’1)i1 ’j1 . (3.51)
i1 ’ j1 i2
46 C. KRATTENTHALER

Then the determinant of M is given by
(N3 )
+2
l
x+i’1 l+i’1
± .
2i ’ 1 2i ’ 1
i=1

Although not completely obvious, the special case x = ’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
n’1
(2i)!2,
det (E2i+2j ) = (3.52)
0¤i,j¤n’1
i=0

where E2k is the (2k)-th (signless) Euler number, de¬ned through the generating function
1/ cos z = ∞ E2k z 2k /(2k)!, and
k=0
n’1
(2i + 1)!2 .
det (E2i+2j+2 ) = (3.53)
0¤i,j¤n’1
i=0

Furthermore, de¬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
n’1
n(n’1)/2
det (Bi+j (x)) = x i!. (3.54)
0¤i,j¤n’1
i=0

Next, there holds
n’1
n(n’1)/2
det (Hi+j (x)) = (’1) i!, (3.55)
0¤i,j¤n’1
i=0
k
where Hm (x) = k≥0 k! (m’2k)! ’ 1 xm’2k is the m-th Hermite polynomial.
m!
2
Finally, the following Hankel determinant evaluations featuring Bernoulli numbers
hold,
n’1
i!6
n
(Bi+j ) = (’1)( )
det , (3.56)
2
(2i)! (2i + 1)!
0¤i,j,¤n’1
i=1

and
n’1
i!3 (i + 1)!3
)1
n+1
(Bi+j+1 ) = (’1)(
det , (3.57)
2
2 (2i + 1)! (2i + 2)!
0¤i,j,¤n’1
i=1

and
n’1
i! (i + 1)!4 (i + 2)!
1 n
det (Bi+j+2 ) = (’1)( ) , (3.58)
2
6 (2i + 2)! (2i + 3)!
0¤i,j,¤n’1
i=1
ADVANCED DETERMINANT CALCULUS 47

and
n’1
(2i)! (2i + 1)!4 (2i + 2)!
det (B2i+2j+2 ) = , (3.59)
(4i + 2)! (4i + 3)!
0¤i,j,¤n’1
i=0

and
n
(2i ’ 1)! (2i)!4 (2i + 1)!
n
det (B2i+2j+4 ) = (’1) . (3.60)
(4i)! (4i + 1)!
0¤i,j,¤n’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 Meixner“Pollaczek polynomials (see [81,
Sec. 1.7]), and use an integral representation for Euler numbers, given for example in

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