Related papers: A Bijection between Necklaces and Restricted Multi…
We give bijective results between several variants of lattice paths of length $2n$ (or $2n-2$) and integer compositions of n, all enumerated by the seemingly innocuous formula $4^{n-1}$. These associations lead us to make new connections…
Two words $p$ and $q$ are avoided by the same number of length-$n$ words, for all $n$, precisely when $p$ and $q$ have the same set of border lengths. Previous proofs of this theorem use generating functions but do not provide an explicit…
Let $G_n$ denote the $n^{\rm th}$ Gleason polynomial, whose roots correspond to parameters $c$ such that the critical point $0$ is periodic of exact period $n$ under iteration of $z^2 + c$, and let $\bar{G}_n$ denote the reduction of $G_n$…
We consider the monomial expansion of the $q$-Whittaker and modified Hall-Littlewood polynomialsarising from specialization of the modified Macdonald polynomial. The two combinatorial formulas for the latter due to Haglund, Haiman, and…
A necklace can be considered as a cyclic list of $n$ red and $n$ blue beads in an arbitrary order, and the goal is to fold it into two and find a large cross-free matching of pairs of beads of different colors. We give a counterexample for…
In a recent paper by Kitaev and Remmel, several formulas for the number of words of length n avoiding some generalized patterns were established. Each time the obtained function of n had been found in Sloane's Encyclopedia as the number of…
It is known that, when $n$ is even, the number of permutations of $\{1,2,\dots,n\}$ all of whose cycles have odd length equals the number of those all of whose cycles have even length. Adin, Heged\H{u}s and Roichman recently found a…
Inspired by the Stochastic processes described by the Feller Coupling and Chinese Restaurant Processes, we create four different bijections from words in the set $[1]\times [2] \times\cdot \times[n]$ to $S_n$. We then compose these maps…
We address two variants of the classical necklace counting problem from enumerative combinatorics. In both cases, we fix a finite group $\mathcal{G}$ and a positive integer $n$. In the first variant, we count the ``identity-product…
In this note we introduce several instructive examples of bijections found between several different combinatorially defined sequences of sets. Each sequence has cardinalities given by the Catalan numbers. Our results answer some questions…
For the double of a quiver, the works of Ginzburg, Bocklandt-Le Bruyn and Schedler show that its closed paths, called the necklaces, have a natural Lie bialgebra structure. Schedler also constructed,in [Int. Math. Res. Notices, 2005 (12),…
We construct weight-preserving bijections between column strict shifted plane partitions with one row and alternating sign trapezoids with exactly one column in the left half that sums to $1$. Amongst other things, they relate the number of…
Kerov, Kirillov, and Reshetikhin defined a bijection between highest weight vectors in the crystal graph of a tensor power of the vector representation, and combinatorial objects called rigged configurations, for type $A^{(1)}_n$. We define…
The connected components of $\mathcal{M}_{0,n}(\mathbb{R})$ are in bijection with the $(n-1)!/2$ dihedral orderings of $[n]$. They are all isomorphic. We construct monomial maps between them, and use these maps to prove a conjecture of…
The number of regions of the type A_{n-1} Shi arrangement in R^n is counted by the intrinsically beautiful formula (n+1)^{n-1}. First proved by Shi, this result motivated Pak and Stanley as well as Athanasiadis and Linusson to provide…
A bijection is presented between (1): partitions with conditions $f_j+f_{j+1}\leq k-1$ and $ f_1\leq i-1$, where $f_j$ is the frequency of the part $j$ in the partition, and (2): sets of $k-1$ ordered partitions $(n^{(1)}, n^{(2)}, ...,…
We introduce a bijection between inequivalent minimal factorizations of the n-cycle (1 2 ... n) into a product of smaller cycles of given length, on one side, and trees of a certain structure on the other. We use this bijection to count the…
A semiorder is a partially ordered set $P$ with two certain forbidden induced subposets. This paper establishes a bijection between $n$-element semiorders of length $H$ and $(n+1)$-node ordered trees of height $H+1$. This bijection…
We describe a combinatorial approach for investigating properties of rational numbers. The overall approach rests on structural bijections between rational numbers and familiar combinatorial objects, namely rooted trees. We emphasize that…
A necklace is an equivalence class of words of length $n$ over an alphabet under the cyclic shift (rotation) operation. As a classical object, there have been many algorithmic results for key operations on necklaces, including counting,…