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