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Differential posets. (English) Zbl 0658.05006
The author has introduced a class of partially ordered sets, called differential posets, with many remarkable combinatorial and algebraic properties. The problem is concerned with counting of saturated chains, $$x_ 1<x_ 2<...<x_ k$$ or of “Hasse walks” $$x_ 1,x_ 2,...,x_ k$$ with either $$x_{i+1}$$ covers $$x_ i$$ or $$x_ i$$ covers $$x_{i+1}$$, $$1\leq i\leq k-1$$ with various properties. A basic tool in the theory of differential posets P is the use of two adjoint linear transformations U and D of vector space of linear combinations of elements of P. If $$x\in P$$ then $$Ux$$ (respectively, $$Dx$$) is the sum of all elements covering $$x$$ (respectively, which $$x$$ covers). A fundamental property of $$U$$ and $$D$$ is the commutative rule $$DU-UD=rI$$ for some positive integer r. Thus differential posets may be regarded as yielding a representation of the “Weyl algebra” generated by $$U$$ and $$D/r$$. The spectrum of the operator $$UD$$ and its eigenvectors are computed and the result is extended to more general functions of $$U$$ and $$D$$. The spectrum of $$UD$$ is closely related to the spectrum of the adjacency matrix of certain finite graphs associated with differential posets.
An example of a differential poset is Young’s lattice Y, first studied by G. Kreweras. It is defined as the set of all partitions of all nonnegative integers n ordered by inclusion of Young diagrams. Thus if $$\lambda =(\lambda_ 1\geq \lambda_ 2\geq...)$$ and $$\mu =(\mu_ 1,\mu_ 2,...)$$ are parititions, $$(\lambda_ 1\geq \lambda_ 2\geq..$$. and $$\mu_ 1\geq \mu_ 2\geq...)$$, then $$\mu\leq \lambda$$ in Y if and only if $$\mu_ i\leq \lambda_ i$$ for all i. Young’s lattice is locally finite distributive lattice. In fact it is the lattice $$J_ f({\mathbb{N}}^ 2)$$ of finite order ideals of the poset $${\mathbb{N}}^ 2$$. If $$\lambda\in Y$$ is a partition of n, it is indicated by $$| \lambda | =n$$ or $$\lambda\vdash n$$. Young’s lattice is graded with rank function $$\rho$$ given by $$\rho(\lambda)=| \lambda |$$. Many remarkable enumerative properties of Y are consequences of the theory of symmetric functions, the representation theory of the symmetric group, the complex general linear group, and the Robinson-Schensted correspondence. A standard Young tableau (SYT) of shape $$\lambda$$ may be identified with a Saturated Chain $$\phi =\lambda^ 0\subset \lambda^ 1\subset...\subset \lambda^ n$$ of partitions from $$\phi$$ to $$\lambda$$. If $$f^{\lambda}$$ denotes the number SYT of shape $$\lambda$$, then $$\sum_{\lambda \vdash n}(f^{\lambda})^ 2=n!$$ asserts that the number of sequences (or Hasse walks) $$\phi =\lambda^ 0<\lambda^ 1<...<\lambda^ n>\mu^{n- 1}>...>\mu^ 0=\phi,$$ where $$\lambda^ i$$ and $$\mu^ i$$ are partitions of i, is equal to $$n!$$.
If P is a graded poset then $$\rho$$ denotes its rank function, i.e., if $$x\in P$$ then $$\rho(x)$$ is the length $$\ell$$ of the longest chain $$x_ 0<x_ 1<...<x_{\ell}=x$$ in P with top element x. Write $$P_ i=\{x\in P:\quad \rho (x)=i\}$$ so $$P=P_ 0\dot\cup P_ 1 \dot\cup...$$ (disjoint union). The counting of chains in partially ordered sets is a well developed subject with many applications in enumerative combinatorics, probability theory and statistical mechanics besides other areas.
Reviewer: M.Cheema

##### MSC:
 05A15 Exact enumeration problems, generating functions 06A06 Partial orders, general 05A17 Combinatorial aspects of partitions of integers 20C30 Representations of finite symmetric groups
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