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On a q-analogue of the Zeta polynomial of posets

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arxiv 2402.11979 v2 pith:LZLGP4WV submitted 2024-02-19 math.CO

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keywords polynomialq-analoguecoefficientszetaalwaysbehaviourchainscharacteristic
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We introduce a q-analogue of the classical Zeta polynomial of finite partially ordered sets, as a polynomial in one variable x with coefficients depending on the indeterminate q. We prove some properties of this polynomial invariant, including its behaviour with respect to duality, product and disjoint union. The leading term is a q-analogue of the number of maximal chains, but not always with non-negative coefficients. The value at q=0 turns out to be essentially the characteristic polynomial.

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