In this paper we enumerate the number of (k, r)-Fuss-Schröder paths of type λ. Y. Park and S. Kim studied small Schröder paths with type λ. Generalizing the results to small (k, r)-Fuss-Schröder paths with type λ, we give a combinatorial interpretation for the number of small (k, r)-Fuss-Schröder paths of type λ by using Chung-Feller style. We also give two sets of sparse noncrossing partitions...

A difference graph is a bipartite graph G = (X, Y: E) such that all the neighborhoods of the vertices of X are comparable by inclusion. We enumerate labeled and unlabeled difference graphs with or without a bipartition of the vertices into two stable sets. The labeled enumerations are expressed in terms of combinatorial numbers related to the Stirling numbers of the second kind.

As the extension of the previous work by Ciucu and the present authors (J. Combin. Theory Ser. A 112(2005) 105–116), this paper considers the problem of enumeration of spanning trees of weighted graphs with an involution which allows fixed points. We show that if G is a weighted graph with an involution, then the sum of weights of spanning trees of G can be expressed in terms of the product of ...

We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then freely placing the structure of a connected positroid on each of the blocks of the partition. This structural result yields several combinatori...

We study the polynomials which enumerate the permutations π = (π1, π2, . . . , πn) of the elements 1.2, . . . , n with the condition π1 < π2 < . . . < πn−m(or π1 > π2 > . . . > πn−m) and prescribed up-down points n−m,n−m+1, . . . , n− 1 in view of an important role of these polynomials in theory of enumeration the permutations with prescribed up-down structure similar to the role of the binomia...

The cyclic sieving phenomenon is defined for generating functions of a set affording a cyclic group action, generalizing Stembridge’s q 1⁄4 1 phenomenon. The phenomenon is shown to appear in various situations, involving q-binomial coefficients, Pólya–Redfield theory, polygon dissections, noncrossing partitions, finite reflection groups, and some finite field q-analogues. r 2004 Elsevier Inc. A...

Pblya's enumeration theorem is generalized in the following way. We have sets R and D, and a group G acting (by means of representations) on R and D simultaneously. This induces an equivalence relation in the set of all mappings (or of all one-to-one mappings) of R into D. The number of equivalence classes is determined for both cases. The example of types of mappings of a set into itself is tr...

We enumerate all positive definite ternary quadratic forms over number fields with class number at most 2. This is done by constructing all definite quaternion orders of type number at most 2 over number fields. Finally, we list all definite quaternion orders of ideal class number 1 or 2.

A leaf of a plane tree is called an old leaf if it is the leftmost child of its parent, and it is called a young leaf otherwise. In this paper we enumerate plane trees with a given number of old leaves and young leaves. The formula is obtained combinatorially by presenting two bijections between plane trees and 2-Motzkin paths which map young leaves to red horizontal steps, and old leaves to up...