A321981 Row n gives the chromatic symmetric function of the n-girder, expanded in terms of elementary symmetric functions and ordered by Heinz number.

1, 2, 0, 6, 0, 0, 16, 0, 2, 0, 0, 40, 12, 2, 0, 0, 0, 0, 96, 16, 44, 6, 0, 0, 0, 0, 0, 0, 0, 224, 136, 66, 52, 2, 4, 0, 2, 0, 0, 0, 0, 0, 0, 0, 512, 384, 208, 96, 30, 178, 0, 18, 30, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1152, 1024, 584, 522, 138, 588, 102

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).

The n-girder has n vertices and looks like:

2-4-6- -n

|\|\|\ ... \|

1-3-5- n-1

Conjecture: All terms are nonnegative (verified up to n = 10). This is a special case of Stanley and Stembridge's poset-chain conjecture.

Links

Table of n, a(n) for n=1..73.

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.

Richard P. Stanley and John R. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, Journal of Combinatorial Theory Series A 62-2 (1993), 261-279.

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

Example

Triangle begins:
    1
    2   0
    6   0   0
   16   0   2   0   0
   40  12   2   0   0   0   0
   96  16  44   6   0   0   0   0   0   0   0
  224 136  66  52   2   4   0   2   0   0   0   0   0   0   0
For example, row 6 gives: X_G6 = 96e(6) + 6e(33) + 16e(42) + 44e(51).

Crossrefs

Row sums are A025192.

Cf. A000569, A001187, A006125, A056239, A229048, A240936, A245883, A277203, A321750, A321911, A321918, A321914, A321979, A321980, A321982.

Sequence in context: A180491 A329893 A047918 * A322481 A262886 A138701

Adjacent sequences: A321978 A321979 A321980 * A321982 A321983 A321984

Keyword

nonn,tabf

Author

Gus Wiseman, Nov 23 2018

Status

approved