By Bjorner A., Stanley R.P.
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Extra info for Combinatorial miscellany
Thus i(w) = (λ), the number of parts of λ. We have shown that for a permutation w with i(w) = p and d(w) = q, the shape λ of the corresponding reverse SYT P (and Q) satisfies (λ) = p and λ1 = q. Hence the number An (p, q) of permutations w of 1, 2, . . , n with i(w) = p and d(w) = q is equal to the number of pairs (P, Q) of reverse SYT of the same shape λ, where λ is a partition of n with (λ) = p and λ1 = q. How many such pairs are there? Given the partition λ, the number of choices for P is just f λ , the number of SYT of shape λ.
In other words, in the case when n is even we would like a combinatorial interpretation of the number M (2n) defined by N (2n, 2n) = M (2n)2 , and similarly when n is odd. – ) found a direct combinatorial interpretation of M (2n). In 1996 Mihai Adrian Ciucu (b. 1968) found an even simpler interpretation of M (2n) as the number of domino tilings of a certain region Rn , up to a power of two. , down to two squares in the last two rows. All the rows are left-justified. The board R4 is illustrated in Figure 5.
Motivated by work related to the adsorption of diatomic molecules on a surface and other physical problems, they were led to consider the tiling of a chessboard by dominos (or dimers). More precisely, consider an m × n chessboard B, where at least one of m and n is even. A domino consists of two adjacent squares (where “adjacent” means having an edge in common). The domino can be oriented either horizontally or vertically. Thus a tiling of B by dominos will require exactly mn/2 dominos, since there are mn squares in all, and each domino has two squares.