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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

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