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Enumerative Combinatorics, Second Edition by Richard P. Stanley

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3.12 Zeta Polynomials

Let P be a finite poset. If n ≥ 2, then define Z(P, n) to be the number of multichains t1t2 ≤ ··· ≤ tn − 1 in P. We call Z(P, n) (regarded as a function of n) the zeta polynomial of P. First we justify this nomenclature and collect together some elementary properties of Z(P, n).

3.12.1 Proposition. a. Let bi be the number of chains t1 < t2 < ··· < ti − 1 in P. Then bi+2 = ΔiZ(P, 2), i ≥ 0, where Δ is the finite difference operator. In other words,

image

(3.51)

In particular, Z(P, n) is a polynomial function of n whose degree d is equal to the length of the longest chain of P, and whose leading coefficient is bd+2/d!. Moreover, ...

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