Related papers: Two Examples of Unbalanced Wilf-Equivalence
Vincular or dashed patterns resemble classical patterns except that some of the letters within an occurrence are required to be adjacent. We prove several infinite families of Wilf-equivalences for k-ary words involving vincular patterns…
In 2012 B\'ona showed the rather surprising fact that the cumulative number of occurrences of the classical patterns $231$ and $213$ are the same on the set of permutations avoiding $132$, beside the pattern based statistics $231$ and $213$…
This paper starts the Wilf-classification of mesh patterns of length 2. Although there are initially 1024 patterns to consider we introduce automatic methods to reduce the number of potentially different Wilf-classes to at most 65. By…
Super-strong (elsewhere referred to as strong) Wilf equivalence is a type of Wilf equivalence on words that was introduced by Kitaev et al. in 2009. We provide a necessary and sufficient condition for two permutations in $n$ letters to be…
We provide a geometric condition that guarantees strong Wilf equivalence in the generalized factor order. This provides a powerful tool for proving specific and general Wilf equivalence results, and several such examples are given.
We offer elementary proofs for several results in consecutive pattern containment that were previously demonstrated using ideas from cluster method and analytical combinatorics. Furthermore, we establish new general bounds on the growth…
We consider a large family of equivalence relations on permutations in Sn that generalise those discovered by Knuth in his study of the Robinson-Schensted correspondence. In our most general setting, two permutations are equivalent if one…
Partially ordered patterns (POPs) generalize the classical notion of permutation patterns within the framework of pattern avoidance. Building on recent work by Burstein, Han, Kitaev, and Zhang, which introduced the concept of…
Building off recent work of Garg and Peng, we continue the investigation into classical and consecutive pattern avoidance in rooted forests, resolving some of their conjectures and questions and proving generalizations whenever possible.…
Two permutation classes, the X-class and subpermutations of the increasing oscillation are shown to exhibit an exponential Wilf-collapse. This means that the number of distinct enumerations of principal subclasses of each of these classes…
Starting with an inclusion-exclusion proof of a combinatorial identity, a direct bijection can be produced using recursive subtraction (sometimes with a direct combinatorial description). We apply this method to identities for generalized…
A grid class consists of permutations whose pictorial depiction can be partitioned into increasing and decreasing parts as determined by a given matrix. In this paper, we introduce a method for enumerating cyclic permutations in vector grid…
A partially ordered pattern (abbreviated POP) is a partially ordered set (poset) that generalizes the notion of a pattern when we are not concerned with the relative order of some of its letters. The notion of partially ordered patterns…
We present a class of permutations for which the number of distinctly ordered subsequences of each permutation approaches an almost optimal value as the length of the permutation grows to infinity.
Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize 3-crossings and 3-nestings,…
Let $p\equiv 1 \pmod{4}$ be a prime. Write $t = \prod_{x=1}^{(p-1)/2}x$. Since $t ^2\equiv -1 \pmod{p}$ , we can divide $\{1,2,\ldots,(p-1)/2\}$ into $(p-1)/4$ ordered pairs so that each pair, say $<a,\tilde{a}>$ , satisfies that $t a…
For two particles with different spin, we derive the Bell's inequality. The inequality is investigated for two systems combining spin-1 and 1/2; spin-1/2 and 3/2. We show that for these states Bell's inequality is violated.
The number of even 321-avoiding permutations of length n is equal to the number of odd ones if n is even, and exceeds it by the (n-1)/2th Catalan number otherwise. We present an involution that proves a refinement of this sign-balance…
We consider the problem of counting the number of possible sets of rankings (called ranking patterns) generated by unfolding models of codimension one. We express the ranking patterns as slices of the braid arrangement and show that all…
We consider the question whether the trigonometric system can be equivalent to some rearrangement of the Walsh system in L_p for some p<>2. We show that this question is closely related to a combinatorial problem. This enables us to prove…