**3.12 Zeta Polynomials**

Let *P* be a finite poset. If *n* ≥ 2, then define *Z(P*, *n*) to be the number of multichains *t*_{1} ≤ *t*_{2} ≤ ··· ≤ *t*_{n − 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 b*_{i} *be the number of chains t*_{1} *< t*_{2} *<* ··· *< t*_{i − 1} *in P. Then b*_{i+2} = Δ^{i}*Z(P*, 2*), i* ≥ 0*, where Δ is the finite difference operator. In other words,*

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*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 b*_{d+2}*/d*!*. Moreover, ...*

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