# [seqfan] additions for completeness' sake

Wouter Meeussen wouter.meeussen at telenet.be
Sun Mar 11 14:12:52 CET 2012

```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
A191714,
with separate entries for their main diagonals and for their row sums.

Would this be too much ballast?
Would anyone ever look them up?

Wouter.

```