Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2508.13709

Cyclic sieving phenomena via combinatorics of continued fractions

Bishal Deb
View original source ↗Read source document ↗Export metadata ↓

Abstract

We will exhibit several instances of the cyclic sieving phenomenon involving statistics and involutions on the following combinatorial families of objects: permutations, set partitions, perfect matchings, D-permutations (and its subclasses). Our results will be based on continued fraction identities enumerating these objects. Our instances of cyclic sieving phenomenon for permutations involve the Corteel involution; this was first studied by Adams, Elder, Lafrenière, McNicholas, Striker and Welch (arxiv~2024). We will reprove several of their results using our setting of continued fractions; we will also prove two of their conjectures. Our study of set partitions and perfect matchings will involve the Kasraoui-Zeng involution and the Chen-Deng-Du-Stanley-Yan (CDDSY) involution. Finally, for D-permutations, we will construct a new involution which we call the Genocchi-Corteel involution. The common feature of all of these involutions, other than the CDDSY involution, is that they are constructed via bijections to weighted lattice paths, and that they exchange crossings and nestings on their respective objects.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Bishal Deb. 2025-08-19. Cyclic sieving phenomena via combinatorics of continued fractions. https://arxiv.org/abs/2508.13709

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗