<<

. 3
( 3)



[122, p. 75],

√ ∞ ’1
(2z)2ν
’1
ν+1
E2ν = (’1) dz.

’∞ ’1 cos πz
Slightly di¬erent proofs of (3.52) can be found in [1] and [108, App. A.5], together
with more Hankel determinant evaluations (among which are also (3.56) and (3.58),
respectively). The evaluation (3.54) can be derived by considering Charlier polyno-
mials (see [35] for such a derivation in a special case). The evaluation (3.55) follows
from the fact that Hermite polynomials are moments of slightly shifted Hermite poly-
nomials, as explained in [71]. In fact, the papers [71] and [72] contain more examples
of orthogonal polynomials which are moments, thus in particular implying Hankel de-
terminant evaluations whose entries are Laguerre polynomials, Meixner polynomials,
and Al-Salam“Chihara polynomials. Hankel determinants where the entries are (clas-
sical) orthogonal polynomials are also considered in [77], where they are related to
Wronskians of orthogonal polynomials. In particular, there result Hankel determinant
evaluations with entries being Legendre, ultraspherical, and Laguerre polynomials [77,
(12.3), (14.3), (16.5), § 28], respectively. The reader is also referred to [103], where
illuminating proofs of these identities between Hankel determinants and Wronskians
are given, by using the fact that Hankel determinants can be seen as certain Schur
functions of rectangular shape, and by applying a ˜master identity™ of Turnbull [178,
p. 48] on minors of a matrix. (The evaluations (3.52), (3.55) and (3.56) can be found in
[103] as well, as corollaries to more general results.) Alternative proofs of (3.52), (3.54)
and (3.55) can be found in [141], see also [139] and [140].
Clearly, to prove (3.56)“(3.58) one would proceed in the same way as in Section 2.7.
(Identity (3.58) is in fact the evaluation (2.38) that we derived in Section 2.7.) The
evaluations (3.59) and (3.60) are equivalent to (3.58), because the matrix underlying
the determinant in (3.58) has a checkerboard pattern (recall that Bernoulli numbers
with odd indices are zero, except for B1 ), and therefore decomposes into the prod-
uct of a determinant of the form (3.59) and a determinant of the form (3.60). Very
interestingly, variations of (3.56)“(3.60) arise as normalization constants in statistical
mechanics models, see e.g. [14, (4.36)], [32, (4.19)], and [108, App. A.5]. A common
generalization of (3.56)“(3.58) can be found in [51, Sec. 5]. Strangely enough, it was
needed there in the enumeration of rhombus tilings.
48 C. KRATTENTHALER

In view of Section 2.7, any continued fraction expansion of the form (2.30) gives rise
to a Hankel determinant evaluation. Thus, many more Hankel determinant evaluations
follow e.g. from work by Rogers [151], Stieltjes [171, 172], Flajolet [44], Han, Randri-
anarivony and Zeng [65, 64, 142, 143, 144, 145, 146, 201], Ismail, Masson and Valent
[70, 73] or Milne [113, 114, 115, 116], in particular, evaluations of Hankel determinant
featuring Euler numbers with odd indices (these are given through the generating func-
tion tan z = ∞ E2k+1 z 2k+1/(2k + 1)!), Genocchi numbers, q- and other extensions of
k=0
Catalan, Euler and Genocchi numbers, and coe¬cients in the power series expansion
of Jacobi elliptic functions. Evaluations of the latter type played an important role in
Milne™s recent beautiful results [113, 114] on the number of representations of integers
as sums of m-th powers (see also [108, App. A.5]).
For further evaluations of Hankel determinants, which apparently do not follow from
known results about continued fractions or orthogonal polynomials, see [68, Prop. 14]
and [51, Sec. 4].
Next we state two charming applications of Lemma 16 (see [189]).
Theorem 53. Let x be a nonnegative integer. For any nonnegative integer n there
hold
n+1
x (2)
(xi)!
det S(xi + j, xi) = (3.61)
(xi + j)! 2
0¤i,j¤n

where 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]), and
n+1
x (2)
(xi)!
s(xi + j, xi) = ’
det , (3.62)
(xi + j)! 2
0¤i,j¤n

where s(m, k) is a Stirling number of the ¬rst kind (up to sign, the number of permu-
tations of m elements with exactly k cycles; cf. [166, p. 18]).

Theorem 54. Let Aij denote the number of representations of j as a sum of i squares
of nonnegative integers. Then det0¤i,j¤n (Aij ) = 1 for any nonnegative integer n. The
same is true if “squares” is replaced by “cubes,” etc.

After having seen so many determinants where rows and columns are indexed by
integers, it is time for a change. There are quite a few interesting determinants whose
rows and columns are indexed by (other) combinatorial objects. (Actually, we already
encountered one in Conjecture 49.)
We start by a determinant where rows and columns are indexed by permutations.
Its beautiful evaluation was obtained at roughly the same time by Varchenko [184] and
Zagier [193].
Theorem 55. For any positive integer n there holds
n
inv(σπ’1 )
n
(1 ’ q i(i’1))( i )(i’2)! (n’i+1)! ,
det q = (3.63)
σ,π∈Sn
i=2

where Sn denotes the symmetric group on n elements.
ADVANCED DETERMINANT CALCULUS 49

This determinant evaluation appears in [193] in the study of certain models in in¬nite
statistics. However, as Varchenko et al. [20, 153, 184] show, this determinant evaluation
is in fact just a special instance in a whole series of determinant evaluations. The
latter papers give evaluations of determinants corresponding to certain bilinear forms
associated to hyperplane arrangements and matroids. Some of these bilinear forms
are relevant to the study of hypergeometric functions and the representation theory
of quantum groups (see also [185]). In particular, these results contain analogues of
(3.63) for all ¬nite Coxeter groups as special cases. For other developments related to
Theorem 55 (and di¬erent proofs) see [36, 37, 40, 67], tying the subject also to the
representation theory of the symmetric group, to noncommutative symmetric functions,
and to free Lie algebras, and [109]. For more remarkable determinant evaluations related
to hyperplane arrangements see [39, 182, 183]. For more determinant evaluations related
to hypergeometric functions and quantum groups and algebras, see [175, 176], where
determinants arising in the context of Knizhnik-Zamolodchikov equations are computed.
The results in [20, 153] may be considered as a generalization of the Shapovalov de-
terminant evaluation [159], associated to the Shapovalov form in Lie theory. The latter
has since been extended to Kac“Moody algebras (although not yet in full generality),
see [31].
There is a result similar to Theorem 55 for another prominent permutation statistics,
MacMahon™s major index. (The major index maj(π) is de¬ned as the sum of all positions
of descents in the permutation π, see e.g. [46].)
Theorem 56. For any positive integer n there holds
n
maj(σπ’1 )
(1 ’ q i )n! (i’1)/i.
det q = (3.64)
σ,π∈Sn
i=2

As Jean“Yves Thibon explained to me, this determinant evaluation follows from
results about the descent algebra of the symmetric group given in [95], presented
there in an equivalent form, in terms of noncommutative symmetric functions. For
the details of Thibon™s argument see Appendix C. Also the bivariate determinant
’1 ’1
detσ,π∈Sn xdes(σπ ) q maj(σπ ) seems to possess an interesting factorization.
The next set of determinant evaluations shows determinants where the rows and
columns are indexed by set partitions. In what follows, the set of all partitions of
{1, 2, . . . , n} is denoted by Πn . The number of blocks of a partition π is denoted by
bk(π). A partition π is called noncrossing, if there do not exist i < j < k < l such
that both i and k belong to one block, B1 say, while both j and l belong to another
block which is di¬erent from B1 . The set of all noncrossing partitions of {1, 2, . . . , n}
is denoted by NCn . (For more information about noncrossing partitions see [160].)
Further, poset-theoretic, notations which are needed in the following theorem are:
Given a poset P , the join of two elements x and y in P is denoted by x ∨P y, while
the meet of x and y is denoted by x §P y. The characteristic polynomial of a poset
P is written as χP (q) (that is, if the maximum element of P has rank h and µ is the
ˆ , where ˆ stands for the
h’rank(p)
M¨bius function of P , then χP (q) :=
o p∈P µ(0, p)q 0
minimal element of P ). The symbol χP (q) denotes the reciprocal polynomial q h χP (1/q)
˜

of χP (q). Finally, P is the order-dual of P .
50 C. KRATTENTHALER

Theorem 57. Let n be a positive integer. Then
n
(n)B(n’i)
i
bk(π§Πn γ)
det q = q χΠ— (q)
˜i , (3.65)
π,γ∈Πn
i=1

where B(k) denotes the k-th Bell number (the total number of partitions of a k-element
set; cf. [166, p. 33]). Furthermore,
n
S(n,i)
q bk(π∨Πn γ) =
det q χΠi (q) , (3.66)
π,γ∈Πn
i=1

where 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]). Next,
n
(2n’1’i)
2n’1
= q( n )
n’1
bk(π§NCn γ)
det q χNCi (q)
˜ , (3.67)
π,γ∈NCn
i=1
and
n
(2n’1’i)
2n
1
() n’1
bk(π∨NCn γ)
det q =q χNCi (q) , (3.68)
n+1 n
π,γ∈NCn
i=1
Finally,
√ 2n
i+1
( )
n’1
Ui+1 ( q/2) n n’1’i
2n’1
= q( ) √
bk(π∨Πn γ)
det q , (3.69)
n
qUi’1 ( q/2)
π,γ∈NCn
i=1
j m’j
(2x)m’2j is the m-th Chebyshev polynomials of the
where Um (x) := j≥0 (’1) j
second kind.
The evaluations (3.65)“(3.68) are due to Jackson [75]. The last determinant eval-
uation, (3.69), is the hardest among those. It was proved independently by Dahab
[33] and Tutte [181]. All these determinants are related to the so-called Birkho¬“Lewis
equation from chromatic graph theory (see [33, 180] for more information).
A determinant of somewhat similar type appeared in work by Lickorish [104] on 3-
manifold invariants. Let NCmatch(2n) denote the set of all noncrossing perfect match-
ings of 2n elements. Equivalently, NCmatch(2n) can be considered as the set of all
noncrossing partitions of 2n elements with all blocks containing exactly 2 elements.
Lickorish considered a bilinear form on the linear space spanned by NCmatch(2n). The
corresponding determinant was evaluated by Ko and Smolinsky [80] using an elegant
recursive approach, and independently by Di Francesco [47], whose calculations are
done within the framework of the Temperley“Lieb algebra (see also [49]).
Theorem 58. For ±, β ∈ NCmatch(2n), let c(±, β) denote the number of connected
components when the matchings ± and β are superimposed. Equivalently, c(±, β) =
bk(± ∨Π2n β). For any positive integer n there holds
n
c(±,β)
Ui (q/2)a2n,2i ,
det q = (3.70)
±,β∈NCmatch(2n)
i=1

where Um (q) is the Chebyshev polynomial of the second kind as given in Theorem 57,
and where a2n,2i = c2n,2i ’ c2n,2i+2 with cn,h = (n’h)/2 ’ (n’h)/2’1 .
n n
ADVANCED DETERMINANT CALCULUS 51

Di Francesco [47, Theorem 2] does also provide a generalization to partial matchings,
and in [48] a generalization in an SU(n) setting, the previously mentioned results being
situated in the SU(2) setting. While the derivations in [47] are mostly combinatorial,
the derivations in [48] are based on computations in quotients of type A Hecke algebras.
There is also an interesting determinant evaluation, which comes to my mind, where
rows and columns of the determinant are indexed by integer partitions. It is a result
due to Reinhart [147]. Interestingly, it arose in the analysis of algebraic di¬erential
equations.
In concluding, let me attract your attention to other determinant evaluations which
I like, but which would take too much space to state and introduce properly.
For example, there is a determinant evaluation, conjectured by Good, and proved by
Goulden and Jackson [60], which arose in the calculation of cumulants of a statistic anal-
ogous to Pearson™s chi-squared for a multinomial sample. Their method of derivation
is very combinatorial, in particular making use of generalized ballot sequences.
Determinants arising from certain raising operators of sl(2)-representations are pre-
sented in [136]. As special cases, there result beautiful determinant evaluations where
rows and columns are indexed by integer partitions and the entries are numbers of
standard Young tableaux of skew shapes.
In [84, p. 4] (see also [192]), an interesting mixture of linear algebra and combinatorial
matrix theory yields, as a by-product, the evaluation of the determinant of certain
incidence mappings. There, rows and columns of the relevant matrix are indexed by all
subsets of an n-element set of a ¬xed size.
As a by-product of the analysis of an interesting matrix in quantum information
theory [93, Theorem 6], the evaluation of a determinant of a matrix whose rows and
columns are indexed by all subsets of an n-element set is obtained.
Determinant evaluations of q-hypergeometric functions are used in [177] to compute
q-Selberg integrals.
And last, but not least, let me once more mention the remarkable determinant evalu-
ation, arising in connection with holonomic q-di¬erence equations, due to Aomoto and
Kato [11, Theorem 3], who thus proved a conjecture by Mimachi [118].

Appendix A: A word about guessing
The problem of guessing a formula for the generic element an of a sequence (an )n≥0
out of the ¬rst few elements was present at many places, in particular this is crucial for
a successful application of the “identi¬cation of factors” method (see Section 2.4) or of
LU-factorization (see Section 2.6). Therefore some elaboration on guessing is in order.
First of all, as I already claimed, guessing can be largely automatized. This is due to
the following tools11 :
1. Superseeker, the electronic version of the “Encyclopedia of Integer Sequences”
[162, 161] by Neil Sloane and Simon Plou¬e (see Footnote 2 in the Introduction),
11
In addition, one has to mention Frank Garvan™s qseries [54], which is designed for guessing and
computing within the territory of q-series, q-products, eta and theta functions, and the like. Procedures
like prodmake or qfactor, however, might also be helpful for the evaluation of “q-determinants”. The
package is available from http://www.math.ufl.edu/˜frank/qmaple.html.
52 C. KRATTENTHALER

2. gfun by Bruno Salvy and Paul Zimmermann and Mgfun by Frederic Chyzak (see
Footnote 3 in the Introduction),
3. Rate by the author (see Footnote 4 in the Introduction).
If you send the ¬rst few elements of your sequence to Superseeker then, if it overlaps
with a sequence that is stored there, you will receive information about your sequence
such as where your sequence already appeared in the literature, a formula, generating
function, or a recurrence for your sequence.
The Maple package gfun provides tools for ¬nding a generating function and/or a
recurrence for your sequence. (In fact, Superseeker does also automatically invoke
features from gfun.) Mgfun does the same in a multidimensional setting.
Within the “hypergeometric paradigm,” the most useful is the Mathematica pro-
gram Rate (“Rate!” is German for “Guess!”), respectively its Maple equivalent GUESS.
Roughly speaking, it allows to automatically guess “closed forms”.12 The program
is based on the observation that any “closed form” sequence (an )n≥0 that appears
within the “hypergeometric paradigm” is either given by a rational expression, like
an = n/(n + 1), or the sequence of successive quotients (an+1 /an )n≥0 is given by a ratio-
nal expression, like in the case of central binomial coe¬cients an = 2n , or the sequence
n
of successive quotients of successive quotients ((an+2 /an+1 )/(an+1 /an ))n≥0 is given by
a rational expression, like in the case of the famous sequence of numbers of alternating
sign matrices (cf. the paragraphs following (3.9), and [18, 19, 111, 148, 97, 198, 199]
for information on alternating sign matrices),
n’1
(3i + 1)!
an = , (A.1)
(n + i)!
i=0

etc. Given enough special values, a rational expression is easily found by rational
interpolation.
This is implemented in Rate. Given the ¬rst m terms of a sequence, it takes the
¬rst m ’ 1 terms and applies rational interpolation to these, then it applies rational
interpolation to the successive quotients of these m ’ 1 terms, etc. For each of the
obtained results it is tested if it does also give the m-th term correctly. If it does, then
the corresponding result is added to the output, if it does not, then nothing is added
to the output.
Here is a short demonstration of the Mathematica program Rate. The output shows
guesses for the i0-th element of the sequence.
In[1]:= rate.m
In[2]:= Rate[1,2,3]
Out[2]= {i0}
In[3]:= Rate[2/3,3/4,4/5,5/6]


12
Commonly, by “closed form” (“NICE” in Zeilberger™s “terminology”) one means an expression
which is built by forming products and quotients of factorials. A strong indication that you encounter
a sequence (an )n≥0 for which a “closed form” exists is that the prime factors in the prime factorization
of an do not grow rapidly as n becomes larger. (In fact, they should grow linearly.)
ADVANCED DETERMINANT CALCULUS 53

1 + i0
Out[3]= {------}
2 + i0

Now we try the central binomial coe¬cients:

In[4]:= Rate[1,2,6,20,70]
-1 + i0
2 (-1 + 2 i1)
Out[4]= { -------------}
i1
i1=1

It needs the ¬rst 8 values to guess the formula (A.1) for the numbers of alternating sign
matrices:

In[5]:= Rate[1,2,7,42,429,7436,218348,10850216]
-1 + i0 -1 + i1
3 (2 + 3 i2) (4 + 3 i2)
Out[5]= { 2( -----------------------)}
4 (1 + 2 i2) (3 + 2 i2)
i1=1 i2=1




However, what if we encounter a sequence where all these nice automatic tools fail?
Here are a few hints. First of all, it is not uncommon to encounter a sequence (an )n≥0
which has actually a split de¬nition. For example, it may be the case that the subse-
quence (a2n )n≥0 of even-numbered terms follows a “nice” formula, and that the subse-
quence (a2n+1)n≥0 of odd-numbered terms follows as well a “nice,” but di¬erent, formula.
Then Rate will fail on any number of ¬rst terms of (an )n≥0 , while it will give you some-
thing for su¬ciently many ¬rst terms of (a2n)n≥0 , and it will give you something else
for su¬ciently many ¬rst terms of (a2n+1 )n≥0 .
Most of the subsequent hints apply to a situation where you encounter a sequence
p0 (x), p1(x), p2 (x), . . . of polynomials pn (x) in x for which you want to ¬nd (i.e., guess)
a formula. This is indeed the situation in which you are generally during the guessing
for “identi¬cation of factors,” and also usually when you perform a guessing where a
parameter is involved.
To make things concrete, let us suppose that the ¬rst 10 elements of your sequence
of polynomials are
54 C. KRATTENTHALER



1 1
6 + 31x ’ 15x2 + 20x3 , 12 ’ 58x + 217x2 ’ 98x3 + 35x4 ,
1, 1 + 2x, 1 + x + 3x2 ,
6 12
1 1
20+508x’925x2 +820x3 ’245x4 +42x5 , 120’8042x+20581x2 ’17380x3 +7645x4 ’1518x5 +154x6 ,
20 120
1
1680 + 386012x ’ 958048x2 + 943761x3 ’ 455455x4 + 123123x5 ’ 17017x6 + 1144x7 ,
1680
1
20160’15076944x+40499716x2 ’42247940x3 +23174515x4 ’7234136x5 +1335334x6 ’134420x7 +6435x8 ,
20160
1
181440 + 462101904x ’ 1283316876x2 + 1433031524x3 ’ 853620201x4 + 303063726x5
181440
’ 66245634x6 + 8905416x7 ’ 678249x8 + 24310x9 , . . . (A.2)


You may of course try to guess the coe¬cients of powers of x in these polynomials.
But within the “hypergeometric paradigm” this does usually not work. In particular,
that does not work with the above sequence of polynomials.
A ¬rst very useful idea is to guess through interpolation. (For example, this is what
helped to guess coe¬cients in [43].) What this means is that, for each pn (x) you try to
¬nd enough values of x for which pn (x) appears to be “nice” (the prime factorization
of pn (x) has small prime factors, see Footnote 12). Then you guess these special eval-
uations of pn (x) (by, possibly, using Rate or GUESS), and, by interpolation, are able to
write down a guess for pn (x) itself.
Let us see how this works for our sequence (A.2). A few experiments will convince
you that pn (x), 0 ¤ n ¤ 9 (this is all we have), appears to be “nice” for x = 0, 1, . . . , n.
Furthermore, using Rate or GUESS, you will quickly ¬nd that, apparently, pn (e) = 2n+e e
for e = 0, 1, . . . , n. Therefore, as it also appears to be the case that pn (x) is of degree
n, our sequence of polynomials should be given (using Lagrange interpolation) by
n
2n + e x(x ’ 1) · · · (x ’ e + 1)(x ’ e ’ 1) · · · (x ’ n)
pn (x) = . (A.3)
e(e ’ 1) · · · 1 · (’1) · · · (e ’ n)
e
e=0

Another useful idea is to try to expand your polynomials with respect to a “suitable”
basis. (For example, this is what helped to guess coe¬cients in [30] or [94, e.g., (3.15),
(3.33)].) Now, of course, you do not know beforehand what “suitable” could be in your
situation. Within the “hypergeometric paradigm” candidates for a suitable basis are
always more or less sophisticated shifted factorials. So, let us suppose that we know
that we were working within the “hypergeometric paradigm” when we came across
the example (A.2). Then the simplest possible bases are (x)k , k = 0, 1, . . . , or (’x)k ,
k = 0, 1, . . . . It is just a matter of taste, which of these to try ¬rst. Let us try the
latter. Here are the expansions of p3 (x) and p4 (x) in terms of this basis (actually, in
terms of the equivalent basis x , k = 0, 1, . . . ):
k

6 + 31x ’ 15x2 + 20x3 = 1 + 6 x x x
1
+ 15 + 20 ,
6 1 2 3
12 ’ 58x + 217x2 ’ 98x3 + 35x4 = 1 + 8 x x x x
1
+ 28 + 56 + 70 .
12 1 2 3 4
I do not know how you feel. For me this is enough to guess that, apparently,
n
2n x
pn (x) = .
k k
k=0
ADVANCED DETERMINANT CALCULUS 55

(Although this is not the same expression as in (A.3), it is identical by means of a
3 F2 -transformation due to Thomae, see [55, (3.1.1)]).
As was said before, we do not know beforehand what a “suitable” basis is. Therefore
you are advised to get as much a priori information about your polynomials as possible.
For example, in [28] it was “known” to the authors that the result which they wanted
to guess (before being able to think about a proof) is of the form (NICE PRODUCT) —
(IRREDUCIBLE POLYNOMIAL). (I.e., experiments indicated that.) Moreover, they
knew that their (IRREDUCIBLE POLYNOMIAL), a polynomial in m, pn (m) say,
would have the property pn (’m ’ 2n + 1) = pn (m). Now, if we are asking ourselves
what a “suitable” basis could be that has this property as well, and which is built
in the way of shifted factorials, then the most obvious candidate is (m + n ’ k)2k =
(m + n ’ k)(m + n ’ k + 1) · · · (m + n + k ’ 1), k = 0, 1, . . . . Indeed, it was very easy
to guess the expansion coe¬cients with respect to this basis. (See Theorems 1 and 2
in [28]. The polynomials that I was talking about are represented by the expression in
big parentheses in [28, (1.2)].)


If the above ideas do not help, then I have nothing else to o¬er than to try some,
more or less arbitrary, manipulations. To illustrate what I could possibly mean, let us
again consider an example. In the course of working on [90], I had to guess the result
of a determinant evaluation (which became Theorem 8 in [90]; it is reproduced here
as Theorem 43). Again, the di¬cult part of guessing was to guess the “ugly” part of
the result. As the dimension of the determinant varied, this gave a certain sequence
pn (x, y) of polynomials in two variables, x and y, of which I display p4 (x, y):

In[1]:= VPol[4]

2 3 4 2 3 2
Out[1]= 6 x + 11 x +6x + x + 6 y - 10 x y - 6 x y - 4 x y + 11 y -

2 2 2 3 3 4
> 6xy +6x y +6y -4xy +y

(What I subsequently describe is the actual way in which the expression for pn (x, y) in
terms of the sum on the right-hand side of (3.38) was found.) What caught my eyes was
the part of the polynomial independent of y, x4 + 6x3 + 11x2 + 6x, which I recognized
as (x)4 = x(x + 1)(x + 2)(x + 3). For the fun of it, I subtracted that, just to see what
would happen:
In[2]:= Factor[%-x(x+1)(x+2)(x+3)]
2 3 2 2 2
Out[2]= y (6 - 10 x - 6 x -4x + 11 y - 6 x y + 6 x y+6y -4xy +

3
y)

Of course, a y factors. Okay, let us cancel that:
In[3]:= %/y
2 3 2 2 2 3
Out[3]= 6 - 10 x - 6 x -4x + 11 y - 6 x y + 6 x y+6y -4xy +y
56 C. KRATTENTHALER

One day I had the idea to continue in a “systematic” manner: Let us subtract/add an
appropriate multiple of (x)3 ! Perhaps, “appropriate” in this context is to add 4(x)3 ,
because that does at least cancel the third powers of x:
In[4]:= Factor[%+4x(x+1)(x+2)]
2 2
Out[4]= (1 + y) (6 - 2 x + 6 x + 5 y - 4 x y + y )

I assume that I do not have to comment the rest:
In[5]:= %/(y+1)
2
Out[5]= 6 - 2 x + 6 x +5y-4xy+y
In[6]:= Factor[%-6x(x+1)]
Out[6]= (2 + y) (3 - 4 x + y)
In[7]:= %/(y+2)
Out[7]= 3 - 4 x + y
In[8]:= Factor[%+4x]
Out[8]= 3 + y

What this shows is that

p4 (x, y) = (x)4 ’ 4(x)3 (y)1 + 6(x)2 (y)2 ’ 4(x)1 (y)3 + (y)4 .

No doubt that, at this point, you would have immediately guessed (as I did) that, in
general, we “must” have (compare (3.38))

n
n
(’1)k
pn (x, y) = (x)k (y)n’k .
k
k=0




Appendix B: Turnbull™s polarization of Bazin™s theorem implies most of the
identities in Section 2.2
In this appendix we show that all the determinant lemmas from Section 2.2, with the
exception of Lemmas 8 and 9, follow from the evaluation of a certain determinant of
minors of a given matrix, an observation which I owe to Alain Lascoux. This evaluation,
due to Turnbull [179, p. 505], is a polarized version of a theorem of Bazin [119, II,
pp. 206“208] (see also [102, Sec. 3.1 and 3.4]).
For the statement of Turnbull™s theorem we have to ¬x an n-rowed matrix A, in which
we label the columns, slightly unconventionally, by a2, . . . , am , b21, b31, b32, b41, . . . , bn,n’1 ,
x1, x2, . . . , xn , for some m ≥ n, i.e., A is an n — (n + m ’ 1 + n ) matrix. Finally, let
2
[a, b, c, . . . ] denote the minor formed by concatenating columns a, b, c, . . . of A, in that
order.
ADVANCED DETERMINANT CALCULUS 57

Proposition 59. (Cf. [179, p. 505], [102, Sec. 3.4]). With the notation as explained
above, there holds

det [bj,1, bj,2 , . . . , bj,j’1 , xi , aj+1 , . . . , am ]
1¤i,j¤n
n
= [x1, x2, . . . , xn , an+1 , . . . , am ] [bj,1, bj,2, . . . , bj,j’1 , aj , . . . , am ]. (B.1)
j=2




Now, in order to derive Lemma 3 from (B.1), we choose m = n and for A the matrix

a2 ... an b21 b31 b32 ... bn,n’1 x1 x2 ... xn
« 
1 ... 1 1 1 1 ... 1 1 1 ... 1
¬ ’A2 X2 . . . X n ·
’An ’B2 ’B2 ’B3 ’Bn
... ... X1
¬ ·
¬ (’A2)2 . . . (’An )2 X2 . . . X n · ,
(’B2 )2 (’B2 )2 (’B3 )2 . . . (’Bn )2 2 2 2
X1
¬ ·
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
(’A2 )n’1 . . . (’An )n’1 (’B2 )n’1 (’B2 )n’1 (’B3 )n’1 . . . (’Bn )n’1 X1 n’1 n’1 n’1
X2 . . . Xn

with the unconventional labelling of the columns indicated on top. I.e., column bst is
¬lled with powers of ’Bt+1 , 1 ¤ t < s ¤ n. With this choice of A, all the minors in (B.1)
are Vandermonde determinants. In particular, due to the Vandermonde determinant
evaluation (2.1), we then have for the (i, j)-entry of the determinant in (B.1)

[bj,1, bj,2, . . . , bj,j’1 , xi, aj+1 , . . . , am]
j n
(Bs ’ Bt ) (As ’ At) (At ’ Bs )
=
2¤s<t¤j j+1¤s<t¤n s=2 t=j+1
j
n
— (Xi + As ) (Xi + Bs ),
s=j+1 s=2

which is, up to factors that only depend on the column index j, exactly the (i, j)-entry
of the determinant in (2.8). Thus, Turnbull™s identity (B.1) gives the evaluation (2.8)
immediately, after some obvious simpli¬cation.
In order to derive Lemma 5 from (B.1), we choose m = n and for A the matrix

a2 ... an b21 b31
«
1 ... 1 1 1
¬ ’A2 ’ C/A2 ’An ’ C/An ’B2,1 ’ C/B2,1 ’B3,1 ’ C/B3,1
...
¬
¬ (’A2 ’ C/A2 )2 . . . (’An ’ C/An )2 (’B2,1 ’ C/B2,1 ) (’B3,1 ’ C/B3,1 )2
2
¬
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(’A2 ’ C/A2 )n’1 . . . (’An ’ C/An )n’1 (’B2,1 ’ C/B2,1 )n’1 (’B3,1 ’ C/B3,1 )n’1

b32 ... bn,n’1 x1 ... xn

1 ... 1 1 ... 1
Xn + C/Xn ·
’B3,2 ’ C/B3,2 ’Bn,n’1 ’ C/Bn,n’1
... X1 + C/X1 ... ·
. . . (Xn + C/Xn )2 · .
(’B3,2 ’ C/B3,2 ) . . . (’Bn,n’1 ’ C/Bn,n’1 )
2 2 2
(X1 + C/X1 ) ·
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(’B3,2 ’ C/B3,2 )n’1 . . . (’Bn,n’1 ’ C/Bn,n’1 )n’1 (X1 + C/X1 )n’1 . . . (Xn + C/Xn )n’1
58 C. KRATTENTHALER

(In this display, the ¬rst line contains columns a2, . . . , b31 of A, while the second line
contains the remaining columns.) Again, with this choice of A, all the minors in (B.1)
are Vandermonde determinants. Therefore, by noting that (S + C/S) ’ (T + C/T ) =
(S ’ T )(C/S ’ T )/(’T ), and by writing pj’1 (X) for
j’1
(X + Bj,s )(C/X + Bj,s ), (B.2)
s=1

we have for the (i, j)-entry of the determinant in (B.1)

(Bj,s + C/Bj,s ’ Bj,t ’ C/Bj,t )
[bj,1, bj,2, . . . , bj,j’1 , xi, aj+1 , . . . , am] =
1¤s<t¤j’1
j’1 n
— (As + C/As ’ At ’ C/At ) (At + C/At ’ Bj,s ’ C/Bj,s )
j+1¤s<t¤n s=1 t=j+1
j’1
n
As ) A’1 ’1
— (Xi + As )(C/Xi + pj’1 (Xi ) Bj,s
s
s=1
s=j+1

for the (i, j)-entry of the determinant in (B.1). This is, up to factors which depend
only on the column index j, exactly the (i, j)-entry of the determinant in (2.11). The
polynomials pj’1 (X), j = 1, 2, . . . , n, can indeed be regarded as arbitrary Laurent poly-
nomials satisfying the conditions of Lemma 5, because any Laurent polynomial qj’1 (X)
over the complex numbers of degree at most j ’ 1 and with qj’1 (X) = qj’1 (C/X) can
be written in the form (B.2). Thus, Turnbull™s identity (B.1) implies the evaluation
(2.11) as well.
Similar choices for A are possible in order to derive Lemmas 4, 6 and 7 (which are in
fact just limiting cases of Lemma 5) from Proposition 59.

Appendix C: Jean-Yves Thibon™s proof of Theorem 56
Obviously, the determinant in (3.64) is the determinant of the linear operator
Kn (q) := σ∈Sn q maj σ σ acting on the group algebra C[Sn ] of the symmetric group.
Thus, if we are able to determine all the eigenvalues of this operator, together with
their multiplicities, we will be done. The determinant is then just the product of all
the eigenvalues (with multiplicities).
The operator Kn (q) is also an element of Solomon™s descent algebra (because permu-
tations with the same descent set must necessarily have the same major index). The
descent algebra is canonically isomorphic to the algebra of noncommutative symmetric
functions (see [56, Sec. 5]). It is shown in [95, Prop. 6.3] that, as a noncommutative
symmetric function, Kn (q) is equal to (q; q)n Sn (A/(1 ’ q)), where Sn (B) denotes the
complete (noncommutative) symmetric function of degree n of some alphabet B.
The inverse element of Sn (A/(1 ’ q)) happens to be Sn((1 ’ q)A), i.e., Sn ((1 ’ q)A) —
Sn(A/(1’q)) = Sn(A),13 with — denoting the internal multiplication of noncommutative
symmetric functions (corresponding to the multiplication in the descent algebra). This
n
is seen as follows. As in [95, Sec. 2.1] let us write σ(B; t) = n≥0 Sn (B)t for the

13
By de¬nition of the isomorphism between noncommutative symmetric functions and elements in
the descent algebra, Sn (A) corresponds to the identity element in the descent algebra of Sn .
ADVANCED DETERMINANT CALCULUS 59

generating function for complete symmetric functions of some alphabet B, and »(B; t) =
n
n≥0 Λn (B)t for the generating function for elementary symmetric functions, which
are related by »(B; ’t)σ(B; t) = 1. Then, by [95, Def. 4.7 and Prop. 4.15], we have
σ((1 ’ q)B; 1) = »(B; ’q)σ(B; 1). Let X be the ordered alphabet · · · < q 2 < q < 1, so
that XA = A/(1 ’ q). According to [95, Theorem 4.17], it then follows that
σ((1 ’ q)A; 1) — σ(XA; 1) = σ((1 ’ q)XA; 1) = »(XA; ’q)σ(XA; 1)
= »(XA; ’q)σ(XA; q)σ(A; 1) = σ(A; 1),
since by de¬nition of X, σ(XA; 1) is equal to σ(XA; q)σ(A; 1) (see [95, Def. 6.1]).
Therefore, Sn ((1 ’ q)A) — Sn (XA) = Sn(A), as required.
Hence, we infer that Kn (q) is the inverse of Sn ((1 ’ q)A)/(q; q)n.
The eigenvalues of Sn ((1 ’ q)A) are given in [95, Lemma 5.13]. Their multiplicities
follow from a combination of Theorem 5.14 and Theorem 3.24 in [95], since the con-
struction in Sec. 3.4 of [95] yields idempotents eµ such that the commutative immage
of ±(eµ ) is equal to pµ /zµ . Explicitly, the eigenvalues of Sn ((1 ’ q)A) are i≥1 (1 ’ q µi ),
where µ = (µ1 , µ2 , . . . ) varies through all partitions of n, with corresponding multiplic-
ities n!/zµ , the number of permutations of cycle type µ, i.e., zµ = 1m1 m1! 2m2 m2! · · · ,
where mi is the number of occurences of i in the partition µ, i = 1, 2, . . . . Hence, the
eigenvalues of Kn (q) are (q; q)n/ i≥1 (1 ’ q µi ), with the same multiplicities.
Knowing all the eigenvalues of Kn (q) and their multiplicities explicitly, it is now not
extremely di¬cult to form the product of all these and, after a short calculation, recover
the right-hand side of (3.64).
Acknowledgements
I wish to thank an anonymous referee, Joris Van der Jeugt, Bernard Leclerc, Madan
Lal Mehta, Alf van der Poorten, Volker Strehl, Jean-Yves Thibon, Alexander Varchenko,
and especially Alain Lascoux, for the many useful comments and discussions which
helped to improve the contents of this paper considerably.
60 C. KRATTENTHALER

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¨ ¨
Institut fur Mathematik der Universitat Wien, Strudlhofgasse 4, A-1090 Wien, Aus-
tria.
e-mail: KRATT@Pap.Univie.Ac.At, WWW: http://radon.mat.univie.ac.at/People/kratt

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