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%I #8 Aug 23 2023 10:17:48
%S 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,
%T 0,2,0,0,0,0,0,0,0,0,0,0,0,810,648,822,56,240,870,280,282,120,24,0,
%U 266,232,0,48,0,54,0,48,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0
%N Row n gives the chromatic symmetric function of the n-ladder, expanded in terms of elementary symmetric functions and ordered by Heinz number.
%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C 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).
%C The n-ladder has 2*n vertices and looks like:
%C o-o-o- -o
%C | | | ... |
%C o-o-o- -o
%C Conjecture: All terms are nonnegative (verified up to the 5-ladder).
%H Richard P. Stanley, <a href="http://www-math.mit.edu/~rstan/pubs/pubfiles/100.pdf">A symmetric function generalization of the chromatic polynomial of a graph</a>, Advances in Math. 111 (1995), 166-194.
%H Richard P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/taor.pdf">Graph colorings and related symmetric functions: ideas and applications</a>, Discrete Mathematics 193 (1998), 267-286.
%H Gus Wiseman, <a href="http://arxiv.org/abs/0709.0430">Enumeration of paths and cycles and e-coefficients of incomparability graphs</a>, arXiv:0709.0430 [math.CO], 2007.
%e Triangle begins:
%e 2 0
%e 12 2 0 0 0
%e 54 26 16 0 2 0 0 0 0 0 0
%e 216 120 168 84 0 24 40 32 0 0 2 0 0 [+9 more zeros]
%e For example, row 3 gives: X_L3 = 54e(6) + 26e(42) + 16e(51) + 2e(222).
%Y Row sums are A109808.
%Y Cf. A000569, A001187, A006125, A056239, A229048, A240936, A245883, A277203, A321911, A321918, A321914, A321979, A321980, A321981.
%K nonn,tabf
%O 1,1
%A _Gus Wiseman_, Nov 23 2018