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A321980 Row n gives the chromatic symmetric function of the n-path, expanded in terms of elementary symmetric functions and ordered by Heinz number. 4
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, 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, 0, 0, 9, 7, 19, 27, 0, 31, 10, 9, 21, 0, 58, 16, 12, 9, 0 (list; graph; refs; listen; history; text; internal format)
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).

All terms are nonnegative [Stanley].

LINKS

Table of n, a(n) for n=1..81.

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.

EXAMPLE

Triangle begins:

  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  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  0  0

For example, row 6 gives: X_P6 = 6e(6) + 10e(42) + 4e(51) + 6e(33) + 2e(222) + 4e(321).

CROSSREFS

Row sums are A000079.

Cf. A000569, A001187, A001349, A006125, A056239, A229048, A240936, A245883, A277203, A321911, A321918, A321914, A321979, A321981, A321982.

Sequence in context: A185914 A144257 A257232 * A208544 A208535 A284856

Adjacent sequences:  A321977 A321978 A321979 * A321981 A321982 A321983

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Nov 23 2018

STATUS

approved

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Last modified February 21 15:11 EST 2019. Contains 320374 sequences. (Running on oeis4.)