Counting Paths in Young’s Lattice
نویسنده
چکیده
Young’s lattice is the lattice of partitions of integers, ordered by inclusion of diagrams. Standard Young tableaux can be represented as paths in Young’s lattice that go up by one square at each step, and more general paths in Young’s lattice correspond to more general kinds of tableaux. Using the theory of symmetric functions, in particular Pieri’s rule for multiplying a Schur function by a complete symmetric function, we derive formulas for counting paths in Young’s lattice that go up or down by horizontal or vertical strips. Our results are related to Richard Stanley’s theory of differential posets in the special case of Young’s lattice.
منابع مشابه
Kazhdan-lusztig and R-polynomials, Young’s Lattice, and Dyck Partitions
We give explicit combinatorial product formulas for the maximal parabolic Kazhdan-Lusztig and R-polynomials of the symmetric group. These formulas imply that these polynomials are combinatorial invariants, and that the KazhdanLusztig ones are nonnegative. The combinatorial formulas are most naturally stated in terms of Young’s lattice, and the one for the Kazhdan-Lusztig polynomials depends on ...
متن کاملCounting Lattice Paths
Counting lattice paths Maciej Dziemiańczuk A lattice path is a finite sequence of points p0, p1, . . . , pn in Z × Z, and a step of the path is the difference between two of its consecutive points, i.e., pi−pi−1. In this thesis, we consider lattice paths running between two fixed points and for which the set of allowable steps contains the vertical step (0,−1) and some number (possibly infinite...
متن کاملCounting Lattice Paths by Gessel Pairs
We count a large class of lattice paths by using factorizations of free monoids. Besides the classical lattice paths counting problems related to Catalan numbers, we give a new approach to the problem of counting walks on the slit plane (walks avoid a half line) that was first solved by Bousquet-Mélou and Schaeffer. We also solve a problem about walks in the half plane avoiding a half line by s...
متن کاملCounting intersecting weighted pairs of lattice paths using transforms of operators
Transforms of linear operators on bivariate generating functions can be used for constructing explicte solutions to certain generalized q di¤erence equations. The method is applied to counting intersecting pairs of lattice paths with weighted turns, a re nement of the q-Narayana numbers.
متن کاملA Representation Theorem for Holonomic Sequences Based on Counting Lattice Paths
Using a theorem of N. Chomsky and M. Schützenberger one can characterize sequences of integers which satisfy linear recurrence relations with constant coefficients (C-finite sequences) as differences of two sequences counting words in regular languages. We prove an analog for P-recursive (holonomic) sequences in terms of counting certain lattice paths.
متن کامل