Duality of Graded Graphs

AbstractA graph is said to be graded if its vertices are divided into levels numbered by integers, so that the endpoints of any edge lie on consecutive levels. Discrete modular lattices and rooted trees are among the typical examples. The following three types of problems are of interest to us:(1) path counting in graded graphs, and related combinatorial identities;(2) bijective proofs of these identities;(3) design and analysis of algorithms establishing corresponding bijections.This article is devoted to (1); its sequel [7] is concerned with the problems (2)–(3). A simplified treatment of some of these results can be found in [8].In this article, R.P. Stanley's [26, 27] linear-algebraic approach to (1) is extended to cover a wide range of graded graphs. The main idea is to consider pairs of graded graphs with a common set of vertices and common rank function. Such graphs are said to be dual if the associated linear operators satisfy a certain commutation relation (e.g., the “Heisenberg” one). The algebraic consequences of these relations are then interpreted as combinatorial identities. (This idea is also implicit in [27].)[7] contains applications to various examples of graded graphs, including the Young, Fibonacci, Young-Fibonacci and Pascal lattices, the graph of shifted shapes, the r-nary trees, the Schensted graph, the lattice of finite binary trees, etc. Many enumerative identities (both known and unknown) are obtained. Abstract of [7]. These identities can also be derived in a purely combinatorial way by generalizing the Robinson-Schensted correspondence to the class of graphs under consideration (cf. [5]). The same tools can be applied to permutation enumeration, including involution counting and rook polynomials for Ferrers boards. The bijective correspondences mentioned above are naturally constructed by Schensted-type algorithms. A general approach to these constructions is given. As particular cases we rederive the classical algorithm of Robinson, Schensted, and Knuth [20, 12, 21], the Sagan-Worley [17, 32] and Haiman [11] algorithms, the algorithm for the Young-Fibonacci graph [5, 15], and others. Several new applications are given.