A321982 Row n gives the chromatic symmetric function of the n-ladder, expanded in terms of elementary symmetric functions and ordered by Heinz number.

2, 0, 12, 2, 0, 0, 0, 54, 26, 16, 0, 2, 0, 0, 0, 0, 0, 0, 216, 120, 168, 84, 0, 24, 40, 32, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 810, 648, 822, 56, 240, 870, 280, 282, 120, 24, 0, 266, 232, 0, 48, 0, 54, 0, 48, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Nov 23 2018

Keywords

• nonn
• tabf

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-ladder has 2*n vertices and looks like:

o-o-o- -o

| | | ... |

o-o-o- -o

Conjecture: All terms are nonnegative (verified up to the 5-ladder).

Examples

			Triangle begins:
    2   0
   12   2   0   0   0
   54  26  16   0   2   0   0   0   0   0   0
  216 120 168  84   0  24  40  32   0   0   2   0   0   [+9 more zeros]
For example, row 3 gives: X_L3 = 54e(6) + 26e(42) + 16e(51) + 2e(222).
		

Links

• 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.

Crossrefs

Row sums are A109808.

Cf. A000569, A001187, A006125, A056239, A229048, A240936, A245883, A277203, A321911, A321918, A321914, A321979, A321980, A321981.