Standard Young tableaux and colored Motzkin paths

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 $2d+1$ rows and the set of SYTs with at most 2d rows.     

The skew, relative, and classical derangements

In his popular combinatories text, Brualdi elucidates the principle of inclusion and exclusion with the classical and the relative derangements. Eventually, the two kinds of derangements are linked up via an algebraic relationship from the parallel use of the principle of inclusion and exclusion. We introduce the notion of skew derangements and relate them to relative derangements and the classical derangements by a purely combinatorial correspondence. Moreover, with the aid of our bijection we easily generalize the relative derangements, obtaining a binomial-type formula for the number of such generalized relative derangements on n elements in terms of the classical derangement number.     

A generalized enumeration of labeled trees and reverse Prüfer algorithm

A leader of a tree T on [n] is a vertex which has no smaller descendants in T. Gessel and Seo showed that

     

A direct bijection between descending plane partitions with no special parts and permutation matrices

We present a direct bijection between descending plane partitions with no special parts and permutation matrices. This bijection has the desirable property that the number of parts of the descending plane partition corresponds to the inversion number of the permutation. Additionally, the number of maximum parts in the descending plane partition corresponds to the position of the one in the last column of the permutation matrix. We also discuss the possible extension of this approach to finding a bijection between descending plane partitions and alternating sign matrices.     

Non-commutative extensions of the MacMahon Master Theorem

We present several non-commutative extensions of the MacMahon Master Theorem, further extending the results of Cartier-Foata and Garoufalidis-Lê-Zeilberger. The proofs are combinatorial and new even in the classical cases. We also give applications to the β-extension and Krattenthaler-Schlosser's q-analogue.     

A simple bijection between a subclass of 2-binary trees and ternary trees

A bijection between binary trees, with nodes labelled black or white, and a black node never having a white right child, and ternary trees is given. It is simpler than another bijection that has recently appeared in this journal.