A321980 - OEIS

A321980

Row n gives the chromatic symmetric function of the n-path, expanded in terms of elementary symmetric functions and ordered by Heinz number.

1, 2, 0, 3, 1, 0, 4, 2, 2, 0, 0, 5, 3, 7, 1, 0, 0, 0, 6, 10, 4, 6, 2, 0, 4, 0, 0, 0, 0, 7, 5, 13, 17, 6, 0, 11, 4, 1, 0, 0, 0, 0, 0, 0, 8, 6, 16, 12, 0, 22, 16, 8, 12, 20, 2, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 9, 7, 19, 27, 0, 31, 10, 9, 21, 0, 58, 16, 12, 9, 0

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OFFSET

1,2

COMMENTS

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The chromatic symmetric function is given by X_G = Sum_p m(t(p)) where the sum is over all stable partitions of G, t(p) is the integer partition whose parts are the block-sizes of p, and m is augmented monomial symmetric functions (see A321895).

All terms are nonnegative [Stanley].

LINKS

Table of n, a(n) for n=1..81.

Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.

Richard P. Stanley, Graph colorings and related symmetric functions: ideas and applications, Discrete Mathematics 193 (1998), 267-286.

Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.

EXAMPLE

Triangle begins:

2 0

3 1 0

4 2 2 0 0

5 3 7 1 0 0 0