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By Krattenthaler C.

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The most objective of those lectures is first to in brief survey the basic con­ nection among the illustration idea of the symmetric workforce Sn and the speculation of symmetric features and moment to teach how combinatorial tools that come up clearly within the thought of symmetric features result in effective algorithms to specific a number of prod­ ucts of representations of Sn when it comes to sums of irreducible representations.

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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. ) 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].

2) counts as well rhombus tilings in a hexagon with side lengths a, b, n, a, b, n. 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 indeterminates. Then there holds q jLi det 1≤i,j≤n Li + A − j Li + j n =q n i=1 iLi i=1 q [Li + A − n]q !

I−b . 49) 0 Then (i) ∆(x; b, c) = 0 if b is even and c is odd; (ii) if any of these conditions does not hold, then b−c ∆(x; b, c) = (−1)c 2c c × i=1 x 1 2 − b 2 i + 12 − (i)c i=1 + c+i 2 c+i + 2 b 2 c b−c+ i/2 − (c+i)/2 b−c+ i/2 − (c+i)/2 x+ 1 2 − b−c+i 2 b + 2 (b+i)/2 − (b−c+i)/2 b−c+i 2 (b+i)/2 − (b−c+i)/2 . 50) The proof of this result in [94] could be called “heavy”. It proceeded by “identification of factors”. 49) that allowed the attack on this special case of the conjecture of Bombieri, Hunt and van der Poorten.

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