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

%I A321981 #7 Aug 23 2023 08:35:38
%S A321981 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,
%T A321981 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,
%U A321981 0,0,0,0,0,0,0,0,0,0,1152,1024,584,522,138,588,102
%N A321981 Row n gives the chromatic symmetric function of the n-girder, expanded in terms of elementary symmetric functions and ordered by Heinz number.
%C A321981 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A321981 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 A321981 The n-girder has n vertices and looks like:
%C A321981   2-4-6-     -n
%C A321981   |\|\|\ ... \|
%C A321981   1-3-5-     n-1
%C A321981 Conjecture: All terms are nonnegative (verified up to n = 10). This is a special case of Stanley and Stembridge's poset-chain conjecture.
%H A321981 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 A321981 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 A321981 Richard P. Stanley and John R. Stembridge, <a href="https://doi.org/10.1016/0097-3165(93)90048-D">On immanants of Jacobi-Trudi matrices and permutations with restricted position</a>, Journal of Combinatorial Theory Series A 62-2 (1993), 261-279.
%H A321981 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 A321981 Triangle begins:
%e A321981     1
%e A321981     2   0
%e A321981     6   0   0
%e A321981    16   0   2   0   0
%e A321981    40  12   2   0   0   0   0
%e A321981    96  16  44   6   0   0   0   0   0   0   0
%e A321981   224 136  66  52   2   4   0   2   0   0   0   0   0   0   0
%e A321981 For example, row 6 gives: X_G6 = 96e(6) + 6e(33) + 16e(42) + 44e(51).
%Y A321981 Row sums are A025192.
%Y A321981 Cf. A000569, A001187, A006125, A056239, A229048, A240936, A245883, A277203, A321750, A321911, A321918, A321914, A321979, A321980, A321982.
%K A321981 nonn,tabf
%O A321981 1,2
%A A321981 _Gus Wiseman_, Nov 23 2018