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A321981
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Row n gives the chromatic symmetric function of the n-girder, expanded in terms of elementary symmetric functions and ordered by Heinz number.
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4
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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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1152, 1024, 584, 522, 138, 588, 102
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OFFSET
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1,2
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COMMENTS
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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-girder has n vertices and looks like:
2-4-6- -n
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1-3-5- n-1
Conjecture: All terms are nonnegative (verified up to n = 10). This is a special case of Stanley and Stembridge's poset-chain conjecture.
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LINKS
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EXAMPLE
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Triangle begins:
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
224 136 66 52 2 4 0 2 0 0 0 0 0 0 0
For example, row 6 gives: X_G6 = 96e(6) + 6e(33) + 16e(42) + 44e(51).
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CROSSREFS
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Cf. A000569, A001187, A006125, A056239, A229048, A240936, A245883, A277203, A321750, A321911, A321918, A321914, A321979, A321980, A321982.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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