Differential Po Sets

In this paper we introduce a class of partially ordered sets, called differential posets, with many remarkable combinatorial and algebraic properties. (Terminology from lattice theory and the theory of partially ordered sets not explained here may be found in [St6].) The combinatorial properties of differential posets are concerned with the counting of saturated chains Xl < x2 < ... < x k or of "Hasse walks" Xl ,x2 ' •.• ,xk (i.e., for 1 :S i :S k 1, either X i+ l covers Xi or Xi covers X i+ l ) with various properties. The counting of chains in partially ordered sets is a well-developed subject with many applications both within and outside of combinatorics. For an introduction to this area, see [St6], especially §§3.S, 3.11, 3.12, 3.13, 3.1S, 3.16, 4.S, and many of the exercises in Chapters 3 and 4. The counting of Hasse walks in a po set is a special case of the counting of walks in a graph. This is the basis for the "transfer-matrix method" [St6, §4.7] of enumerative combinatorics, with many applications to probability theory (Markov chains in particular), statistical mechanics, and other areas. See [C-D-S] for additional information. A basic tool in the theory of differential posets P is the use of two adjoint linear transformations U and D on the vector space of linear combinations of elements of P. If X E P then Ux (respectively, Dx) is the sum of all elements covering X (respectively, which X covers). Such linear transformations have appeared in many contexts, but rarely does one have such explicit information as for differential posets. Linear transformations identical or similar to U and/or D appear, for instance, in [P, StS, St7] in order to obtain structural information about P; for differential posets the corresponding result is our Corollary 4.4. One may also regard U and D as instances of the finite Radon transform (e.g., [Ku]). For differential po sets a fundamental property of U and D is the commutation rule DU UD = rI for some positive integer r (see Theorem 2.2). Thus differential posets may be regarded as affording a representation of the "Weyl