Some Triangulated Surfaces without Balanced Splitting

Let G be the graph of a triangulated surface $\Sigma$ of genus $g\geq 2$. A cycle of G is splitting if it cuts $\Sigma$ into two components, neither of which is homeomorphic to a disk. A splitting cycle has type k if the corresponding components have genera k and g-k. It was conjectured that G contains a splitting cycle (Barnette '1982). We confirm this conjecture for an infinite family of triangulations by complete graphs but give counter-examples to a stronger conjecture (Mohar and Thomassen '2001) claiming that G should contain splitting cycles of every possible type.Comment: 15 pages, 7 figure

Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions

Using the bijection between partitions and vacillating tableaux, we establish a correspondence between pairs of noncrossing free Dyck paths of length $2n$ and noncrossing partitions of $[2n+1]$ with $n+1$ blocks. In terms of the number of up steps at odd positions, we find a characterization of Dyck paths constructed from pairs of noncrossing free Dyck paths by using the Labelle merging algorithm.Comment: 9 pages, 5 figures, revised version, to appear in Discrete Mathematic

An involution on Dyck paths and its consequences

AbstractAn involution is introduced in the set of all Dyck paths of semilength n from which one re-obtains easily the equidistribution of the parameters ‘number of valleys’ and ‘number of doublerises’ and also the equidistribution of the parameters ‘height of the first peak’ and ‘number of returns’

Permutations with forbidden subsequences and a generalized Schröder number

AbstractUsing the technique of generating trees, we prove that there are exactly 10 classes of pattern avoiding permutations enumerated by the large Schröder numbers. For each integer, m⩾1, a sequence which generalizes the Schröder and Catalan numbers is shown to enumerate m+22 classes of pattern avoiding permutations. Combinatorial interpretations in terms of binary trees and polyominoes and a generating function for these sequences are given

Enumerating indeterminacy

In much of the indeterminate music composed in the 1950s and 60s, the roles of the composer and performer are blurred, the performer having been given control over musical elements previously dictated solely by the composer. Often, decisions must be made by the performer that impact a work’s form and content, the composer’s quiet voice heard only in the directives influencing these decisions. In many cases, these directives lack specificity, allowing for an infinite number of performance possibilities; in some cases, however, composer directives severely restrict that number, permitting it to be discretely counted. To these latter cases we turn our attention, mathematically modeling the composer’s directives to enumerate all possible realizations of certain indeterminate scores. Taking Morton Feldman’s Durations 2 and Karlheinz Stockhausen’s Klavierstück XI as primary examples, we calculate the total number of possible realizations, generalizing each case in order to enumerate the realizations of other works with similar characteristics

Q.954 and A.954, Quickie Problem and Solution

Problem and proof proposed by authors. Another proof, using lattice paths, can be found in Robert A. Sulanke\u27s article, Objects Counted by the Central Delannoy Numbers, The Journal of Integer Sequences, Vol 6, 2003. A proof by polynomials is in Michel Bataille\u27s paper Some Identities about an Old Combinatorial Sum, The Mathematical Gazette, March 2003, pp. 144-8. A slight change in the above proof leads to m ≥ n, a generalization proved by Li Zhou using lattice paths in The Mathematical Gazette