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A recurrence for linear extensions. (English) Zbl 0692.06001
This article deals with the number e(P) of linear extensions of a finite partially ordered set P (the set P does not change). For the formulation of the main theorem the following notion is necessary. Let $$C=\{x_ 0<x_ 1<...<x_ m\}$$ be a saturated chain in P $$(x_{i+1}$$ covers $$x_ i$$ for $$0\leq i<m)$$. Put $$P_ C=P-\{x_ 0\}$$ if $$m=0$$ and $$P_ C=(P-C)\cup \{x_{0,1},x_{1,2},...,x_{m-1,m}\},$$ where $$x_{0,1},...,x_{m-1,m}$$ are new elements. The partial ordering on $$P_ C$$ is added by $$x_{0,1}<...<x_{m-1,m}$$, $$y<x_{i,i+1}$$ if $$y\in P-C$$ and $$y<x_{i+1}$$ in P, $$y>x_{i,i+1}$$ if $$y\in P-C$$ and $$y>x_ i$$ in P.
The authors prove the following three assertions: Theorem. Let $${\mathfrak C}$$ be a set of saturated chains of P such that every maximal chain of P contains exactly one element of $${\mathfrak C}$$. Then $$e(P)=\sum_{C}e(P_ C)$$ (C$$\in {\mathfrak C}).$$
Corollary. Let A be an antichain of P which intersects every maximal chain. Then $$e(P)=\sum_{x}e(P-\{x\})$$ (x$$\in A).$$
Proposition. If P and Q are finite posets with isomorphic comparability graphs, then $$e(P)=e(Q)$$.
Reviewer: L.Skula

##### MSC:
 06A06 Partial orders, general 06A05 Total orders
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##### References:
 [1] B. Dreesen, W. Poguntke, and P. Winkler (1985) Comparability invariance of the fixed point property, Order 2, 269-274. · Zbl 0577.06005 [2] M. Habib (1984) Comparability invariants, Ann. Discrete Math. 23, 371-386. [3] D. Kelly (1986) Invariants of finite comparability graphs, Order 3, 155-158. · Zbl 0616.05059 · doi:10.1007/BF00390105 [4] M. P. Schützenberger (1972) Promotion des morphismes d’ensembles ordonnes, Discrete Math. 2, 73-94. · Zbl 0279.06001 · doi:10.1016/0012-365X(72)90062-3 [5] R. Stanley (1986) Two poset polytopes, Discrete Comput. Geom. 1, 9-23. · Zbl 0595.52008 · doi:10.1007/BF02187680 [6] R. Stanley (1986) Enumerative Combinatorics, Vol. 1, Wadsworth & Brooks/Cole, Pacific Grove, CA. · Zbl 0608.05001
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