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We use the algebra of difference operators to study sums of squares (and other powers) of the characters of the symmetric group, χ λ (µ), when the sum is restricted over shapes, λ, with a fixed number of rows, and for hook shapes, and µ has 'mostly' ones. We prove that such sums are always P-recursive, i.e., satisfy a linear difference equation with(More)
Recall that one of the almost infinitely many definitions of the ubiquitous Catalan Numbers, C n , is as the number of elements of the set of 2n-letter words, w 1. .. w 2n in the alphabet {−1, 1} that add up to zero, and all whose partial sums are non-negative. Let's call this set C n. In the 1924 Toronto ICM, Jacques Touchard [T] announced (and proved) the(More)
In a recent article, we noted (and proved) that the sum of the squares of the characters of the symmetric group, χ λ (µ), over all shapes λ with two rows and n cells and µ = 31 n−3 , equals, surprisingly, to 1/2 of that sum-of-squares taken over all hook shapes with n + 2 cells and with µ = 321 n−3. In the present note, we show that this is only the tip of(More)