[seqfan] Re: additions for completeness' sake
njasloane at gmail.com
Sun Mar 11 14:42:56 CET 2012
Wouter, That's an excellent observation. Please add a comment
to A124577, and add the others as new sequences!
On Sun, Mar 11, 2012 at 9:12 AM, Wouter Meeussen <wouter.meeussen at telenet.be
> A001700 C(2n+1, n+1): number of ways to put n+1 indistinguishable
> balls into 2n+1 distinguishable boxes = number of (n+1)-st degree monomials
> in n+1 variables = number of monotone maps from 1..n+1 to 1..n+1.
> This (Offset 0) sequence thus also counts the monomial symmetric functions
> of (degree=number of variables).
> Now, the monomial symmetric functions are only one of a set of 5, the
> others are
> Power Sum Symmetric Polynomials, Complete Homogeneous Symmetric
> Polynomials, Elementary Symmetric Polynomials and Schur Polynomials:
> So I looked them up, and it turns out only the Power Sum Symm. poly's give
> a hit in OEIS:
> 1, 6, 39, 356, 4055, 57786, 983535, 19520264, 441967518
> A124577: "Define p(alpha) to be the number of H-conjugacy classes where H
> is a Young subgroup of type alpha of the symmetric group S_n. Then a(n) =
> sum p(alpha) where |alpha| = n and alpha has at most n parts."
> without ('direct') mention of symmetric functions.
> no hits for the others:
> Complete Homogeneous Symmetric Polynomials
> 1, 7, 55, 631, 8001, 130453, 2323483, 48916087, 1129559068
> Elementary Symmetric Polynomials
> 1, 5, 37, 405, 5251, 84893, 1556535, 33175957, 785671039
> Schur Polynomials
> 1, 4, 19, 116, 751, 5552, 43219, 366088, 3245311
> though this one is 'hidden' as main diagonal of triangle A191714.
> This suggests looking at these symmetric poly's as triangular tables like
> with separate entries for their main diagonals and for their row sums.
> Would this be too much ballast?
> Would anyone ever look them up?
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