cp's OEIS Frontend

A321980 Row n gives the chromatic symmetric function of the n-path, expanded in terms of elementary symmetric functions and ordered by Heinz number.

%I A321980 #7 Aug 23 2023 08:36:25
%S A321980 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,
%T A321980 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,
%U A321980 0,0,9,7,19,27,0,31,10,9,21,0,58,16,12,9,0
%N A321980 Row n gives the chromatic symmetric function of the n-path, expanded in terms of elementary symmetric functions and ordered by Heinz number.
%C A321980 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A321980 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 A321980 All terms are nonnegative [Stanley].
%H A321980 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 A321980 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 A321980 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 A321980 Triangle begins:
%e A321980   1
%e A321980   2  0
%e A321980   3  1  0
%e A321980   4  2  2  0  0
%e A321980   5  3  7  1  0  0  0
%e A321980   6 10  4  6  2  0  4  0  0  0  0
%e A321980   7  5 13 17  6  0 11  4  1  0  0  0  0  0  0
%e A321980   8  6 16 12  0 22 16  8 12 20  2  0  0  6  0  0  0  0  0  0  0  0
%e A321980 For example, row 6 gives: X_P6 = 6e(6) + 10e(42) + 4e(51) + 6e(33) + 2e(222) + 4e(321).
%Y A321980 Row sums are A000079.
%Y A321980 Cf. A000569, A001187, A001349, A006125, A056239, A229048, A240936, A245883, A277203, A321911, A321918, A321914, A321979, A321981, A321982.
%K A321980 nonn,tabf
%O A321980 1,2
%A A321980 _Gus Wiseman_, Nov 23 2018