Tung Shan Fu

Refined Chung-Feller theorems for lattice paths

In this paper we prove a strengthening of the classical Chung-Feller theorem and a weighted version for Schroder paths. Both results are proved by refined bijections which are developed from the study of Taylor expansions of generating functions. By the same technique, we establish variants of the bijections for Catalan paths of order d and certain families of Motzkin paths. Moreover, we obtain a neat formula for enumerating schroder paths with flaws.

Area of Catalan paths on a checkerboard

It is known that the area of all Catalan paths of length n is equal to 4^n-2n+1n, which coincides with the number of inversions of all 321-avoiding permutations of length n+1. In this paper, a bijection between the two sets is established. Meanwhile, a number of interesting bijective results that pave the way to the required bijection are presented.

Standard Young tableaux and colored Motzkin paths

Abstract In this paper, we propose a notion of colored Motzkin paths and establish a bijection between the n-cell standard Young tableaux (SYT) of bounded height and the colored Motzkin paths of length n. This result not only gives a lattice path interpretation of the standard Young tableaux but also reveals an unexpected intrinsic relation between the set of SYTs with at most 2 d + 1 rows and the set of SYTs with at most 2d rows.

Sign-balance identities of Adin-Roichman type on 321-avoiding alternating permutations

Abstract Adin and Roichman proved a set of refined sign-balance identities on 321-avoiding permutations respecting the last descent of the permutations, which we call the identities of Adin–Roichman type. In this work, we construct a new involution on plane trees that proves refined sign-balance properties on 321-avoiding alternating permutations respecting the first and last entries of the permutations respectively and obtain two sets of identities of Adin–Roichman type.

On Simsun and Double Simsun Permutations Avoiding a Pattern of Length Three

A permutation σ ∈

Lattice paths and generalized cluster complexes

\frak{S}_n

Signed countings of types B and D permutations and t,q-Euler numbers

is simsun if for all k, the subword of σ restricted to {1, . . . , k} does not have three consecutive decreasing elements. The permutation σ is double simsun if both σ and σ−1 are simsun. In this paper, we present a new bijection between simsun permutations and increasing 1-2 trees, and show a number of interesting consequences of this bijection in the enumeration of pattern-avoiding simsun and double simsun permutations. We also enumerate the double simsun permutations that avoid each pattern of length three.

A refined sign-balance of simsun permutations

In this paper we propose a variant of the generalized Schroder paths and generalized Delannoy paths by giving a restriction on the positions of certain steps. This generalization turns out to be reasonable, as attested by the connection with the faces of generalized cluster complexes of types A and B. As a result, we derive Krattenthalers F-triangles for these two types by a combinatorial approach in terms of lattice paths.

A Simple Transformation for Mahonian Statistics on Labelings of Rake Posets

It is a classical result that the parity-balance of the number of weak excedances of all permutations (derangements, respectively) of length

Two refined major-balance identities on 321-avoiding involutions

n