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Hiranya Kishore Dey

Publications and source records attributed to Hiranya Kishore Dey.

At least 19 recordsLinked to original sources
arXiv2026

Group Structure from Subgroup and Cyclic Subgroup Counts

For a finite group \(G\), let \(\sub(G)\) be the number of subgroups of \(G\), let \(\cyc(G)\) be the number of cyclic subgroups, and let \(π(G)\) be the number of distinct prime divisors of \(|G|\). We study the normalized counts \(λ(G)=\sub(G)/2^{π(G)}\) and \(η(G)=\cyc(G)/2^{π(G)}\). We prove that \(η(G)<5/4\) or \(λ(G)<3/2\) implies that \(G\) is cyclic of squarefree order. The inequalities \(η(G)<2\) and \(λ(G)<5/2\) each force all Sylow subgroups to be cyclic, and hence imply metacyclicity. For the subgroup count, we give an exact arithmetic criterion in the parameters of the corresponding \(ZM\)-presentation. We determine all values with \(1<η(G)<2\) and \(1<λ(G)<5/2\), and prove that \(λ(G)<59/8\) or \(η(G)<4\) implies solvability. Both solvability bounds are sharp. We also describe how cyclic direct factors of coprime order affect the two normalized counts.

math.GR↗
arXiv2026

How to pick your team with no size restriction

Settling a problem raised by Eccles in 2015, Narayanan in 2026 considers a two-player game in which two captains alternately select players while the opponent decides to which team each selected player is assigned. Moreover, the two teams are required to have equal cardinalities, and Narayanan proved that the second player has a non-losing strategy. In this paper, we study a natural variant in which the teams are allowed to have different cardinalities, and the winner is determined by comparing the average strengths of the two teams. We show that, in this setting, the parity of the total number of players completely determines which player has a non-losing strategy: the first player has a non-losing strategy when the number of players is even, while the second player has a non-losing strategy when it is odd.

math.CO↗
arXiv2026

Centralizers in finite groups and Domination number of their commuting graphs

The proper commuting graph $\mathcal{C}^{**}(G)$ of a finite group $G$ is the simple graph whose vertices are the noncentral elements of $G$ and two distinct vertices are adjacent if they commute. In this paper, we study the domination number and total domination number of proper commuting graphs of finite groups. We first obtain general bounds for the domination number of proper commuting graphs. For finite nilpotent groups, we exploit a strong product decomposition of commuting graphs to derive exact formulas for the domination number. We further determine the exact domination number and total domination number for proper commuting graphs of several well-known families of finite groups, connecting with the centralizers of those groups.

math.CO↗
arXiv2025

$q$-analogues of Fisher's inequality and oddtown theorem

A classical result in design theory, known as Fisher's inequality, states that if every pair of clubs in a town shares the same number of members, then the number of clubs cannot exceed the number of inhabitants in the town. In this short note, we establish a $q$-analogue of Fisher's inequality. Additionally, we present a $q$-analogue of the oddtown theorem for the case when $q$ is an odd prime power.

math.CO↗
arXiv2024

A descent-excedance correspondence in colored permutation groups

It is well known that descents and excedances are equidistributed in the symmetric group. We show that the descent and excedance enumerators, summed over permutations with a fixed first letter are identical when we perform a simple change of the first letter. We generalize this to type B and other colored permutation groups. We are led to defining descents and excedances through linear orders. With respect to a particular order, when the number of colors is even, we get a result that generalizes the type B results. Lastly, we get a type B counterpart of Conger's result which refines the well known Carlitz identity.

math.CO↗
arXiv2024

Sign-balance of excedances over mod-k-alternating permutations and gamma-positivity

A permutation is called mod-k-alternating if its entries are restricted to having the same remainder as the index, modulo some integer $k \geq 1.$ In this paper, we find the sign-balance for mod-k-alternating permutations with respect to the statistic excedance. Moreover, we study the sign-balance for excedances over mod-k-alternating derangements. The results are obtained by constructing suitable matrices and connecting their determinants with the signed excedance enumeration of mod-k-alternating permutations. As an application of the signed excedance enumeration, we prove that when $n \equiv k \pmod {2k}$, the excedance enumerating polynomials over the even and odd mod-k-alternating permutations, starting with a fixed remainder, are gamma-positive.

math.CO↗
arXiv2024

How large is the character degree sum compared to the character table sum for a finite group?

In 1961, Solomon gave upper and lower bounds for the sum of all the entries in the character table of a finite group in terms of elementary properties of the group. In a different direction, we consider the ratio of the character table sum to the sum of the entries in the first column, also known as the character degree sum, in this work. First, we propose that this ratio is at most two for many natural groups. Secondly, we extend a conjecture of Fields to postulate that this ratio is at least one with equality if and only if the group is abelian. We establish the validity of this property and conjecture for all finite irreducible Coxeter groups. In addition, we prove the conjecture for generalized symmetric groups. The main tool we use is that the sum of a column in the character table of an irreducible Coxeter group (resp. generalized symmetric group) is given by the number of square roots (resp. absolute square roots) of the corresponding conjugacy class representative. As a byproduct of our results, we show that the asymptotics of character table sums is the same as the number of involutions in symmetric, hyperoctahedral and demihyperoctahedral groups. We also derive explicit generating functions for the character table sums for these latter groups as infinite products of continued fractions. In the same spirit, we prove similar generating function formulas for the number of square roots and absolute square roots in $n$ for the generalized symmetric groups $G(r,1,n)$.

math.RT↗
arXiv2024

On the order sequence of a group

This paper provides a bridge between two active areas of research, the spectrum (set of element orders) and the power graph of a finite group. The order sequence of a finite group $G$ is the list of orders of elements of the group, arranged in non-decreasing order. Order sequences of groups of order $n$ are ordered by elementwise domination, forming a partially ordered set. We prove a number of results about this poset, among them the following. M.~Amiri recently proved that the poset has a unique maximal element, corresponding to the cyclic group. We show that the product of orders in a cyclic group of order $n$ is at least $q^{ϕ(n)}$ times as large as the product in any non-cyclic group,where $q$ is the smallest prime divisor of $n$ and $ϕ$ is Euler's function, with a similar result for the sum. The poset of order sequences of abelian groups of order $p^n$ is naturally isomorphic to the (well-studied) poset of partitions of $n$ with its natural partial order. If there exists a non-nilpotent group of order $n$, then there exists such a group whose order sequence is dominated by the order sequence of any nilpotent group of order $n$. There is a product operation on finite ordered sequences, defined by forming all products and sorting them into non-decreasing order. The product of order sequences of groups $G$ and $H$ is the order sequence of a group if and only if $|G|$ and $|H|$ are coprime. The paper concludes with a number of open problems.

math.GR↗
arXiv2023

Combinatorial proof of the log-convexity for the derangements in the Coxeter groups

We provide the combinatorial proofs of the log-convexity for the derangement numbers in the symmetric group $\mathfrak{S}_n$, hyperoctahedral group $\mathfrak{B}_n$, and the demihyperoctahedral group $\mathfrak{D}_n$. We also show that the sequences of the even and odd derangement numbers in $\mathfrak{S}_n$ and $\mathfrak{B}_n$ are log-convex.

math.CO↗
arXiv2023

Distance matrix of enhanced power graphs of finite groups

The enhanced power graph of a group $G$ is the graph $\mathcal{G}_E(G)$ with vertex set $G$ and edge set $ \{(u,v): u, v \in \langle w \rangle,~\mbox{for some}~ w \in G\}$. In this paper, we compute the spectrum of the distance matrix of the enhanced power graph of non-abelian groups of order $pq$, dihedral groups, dicyclic groups, elementary abelian groups $\mathrm{El}(p^n)$ and the non-cyclic abelian groups $\mathrm{El}(p^n)\times\mathrm{El}(q^m)$ and $\mathrm{El}(p^n)\times \mathbb{Z}_m$, where $p$ and $q$ are distinct primes. For the non-cyclic abelian group $\mathrm{El}(p^n)\times \mathrm{El}(q^m)$, we also compute the spectrum of the adjacency matrix of its enhanced power graph and the spectrum of the adjacency and the distance matrix of its power graph.

math.CO↗
arXiv2023

Sharp lower bound on the "number of nodal decomposition" of graphs

Urschel introduced a notion of nodal partitioning to prove an upper bound on the number of nodal decomposition of discrete Laplacian eigenvectors. The result is an analogue to the well-known Courant's nodal domain theorem on continuous Laplacian. In this article, using the same notion of partitioning, we discuss the lower bound (or lack thereof) on the number of nodal decomposition of eigenvectors in the class of all graphs with a fixed number of vertices (however large). This can be treated as a discrete analogue to the results of Stern and Lewy in the continuous Laplacian case.

math.CO↗
arXiv2023

$q$-enumeration of type B and D Eulerian polynomials based on parity of descents

Carlitz and Scoville in 1973 considered a four variable polynomial that enumerates permutations in $\mathfrak{S}_n$ with respect to the parity of its descents and ascents. In recent work, Pan and Zeng proved a $q$-analogue of Carlitz-Scoville's generating function by enumerating permutations with the above four statistice along with the inversion number. Further, they also proved a type B analogue by enumerating signed permutations with respect to the parity of descents and ascents. In this work we prove a $q$-analogue of the type B result of Pan and Zeng by enumerating permutations in $\mathfrak{B}_n$ with the above four statistics and the type B inversion number. We also obtain a $q$-analogue of the generating function for the type B bivariate alternating descent polynomials. We consider a similar five-variable polynomial in the type D Coxeter groups as well and give their egf. Alternating descents for the type D groups were previously also defined by Remmel, but our definition is slightly different. As a by-product of our proofs, we get bivariate $q$-analogues of Hyatt's recurrences for the type B and type D Eulerian polynomials. Further corollaries of our results are some symmetry relations for these polynomials and $q$-analogues of generating functions for snakes of types B and D.

math.CO↗
arXiv2022

The solvability of a finite group by the sum of powers of element orders

We prove a new criterion for the solvability of the finite groups, depending on the function $ψ_k(G)$ which is defined as the sum of $k$-th powers of the element orders of $G$. We show that our result can be used to show the solvability of some groups for which the solvability does not follow from earlier similar kind of results and we emphasize the following: looking at $ψ_k(G)$ for $k>1$ can be useful to get further pieces of information about the group $G$.

math.GR↗
arXiv2022

On Difference of Enhanced Power Graph and Power Graph of a Finite Group

The difference graph $D(G)$ of a finite group $G$ is the difference of enhanced power graph of $G$ and power graph of $G$, with all isolated vertices are removed. In this paper we study the connectedness and perfectness of $D(G)$ with respect to various properties of the underlying group $G$. We also find several connection between the difference graph of $G$ and the Gruenberg-Kegel graph of $G$.

math.CO↗
arXiv2022

The Young matroid: A multiset extension of the Catalan matroid to arbitrary Young diagrams

Introduced by Ardila (J. Combin. Theory Ser. A, 2003), the Catalan matroid is obtained by defining the bases of the matroid using Dyck paths from $(0,0)$ to $(n,n)$. Further research has gone into the topic, with variants like lattice path matroids (introduced by Bonin, de Mier, and Noy (J. Combin. Theory Ser. A, 2003)) and shifted matroids (introduced independently by Klivans (2003), and Ardila) being studied intensively. In this short note, we introduce the Young matroid, an extension of the Catalan matroid, where the bases are defined using the standard Young tableaux of a fixed shape. This extension necessarily involves the consideration of independent multisets and multiset bases.

math.CO↗
arXiv2022

An exact upper bound for the sum of powers of element orders in non-cyclic finite groups

For a finite group $G$, let $ψ(G)$ denote the sum of element orders of $G$. This function was introduced by Amiri, Amiri, and Isaacs in 2009 and they proved that for any finite group $G$ of order $n$, $ψ(G)$ is maximum if and only if $G \simeq \mathbb{Z}_n$ where $\mathbb{Z}_n$ denotes the cyclic group of order $n$. Furthermore, Herzog, Longobardi, and Maj in 2018 proved that if $G$ is non-cyclic, $ψ(G) \leq \frac{7}{11} ψ(\mathbb{Z}_n)$. Amiri and Amiri in 2014 introduced the function $ψ_k(G)$ which is defined as the sum of the $k$-th powers of element orders of $G$ and they showed that for every positive integer $k$, $ψ_k(G)$ is also maximum if and only if $G$ is cyclic. In this paper, we have been able to prove that if $G$ is a non-cyclic group of order $n$, then $ψ_k(G) \leq \frac{1+3.2^k}{1+2.4^k+2^k} ψ_k(\mathbb{Z}_n)$. Setting $k=1$ in our result, we immediately get the result of Herzog et al. as a simple corollary. Besides, a recursive formula for $ψ_k(G)$ is also obtained for finite abelian $p$-groups $G$, using which one can explicitly find out the exact value of $ψ_k(G)$ for finite abelian groups $G$.

math.GR↗
arXiv2022

Determining Number of Kneser Graphs: Exact Values and Improved Bounds

The determining number of a graph $G = (V,E)$ is the minimum cardinality of a set $S\subseteq V$ such that pointwise stabilizer of $S$ under the action of $Aut(G)$ is trivial. In this paper, we provide some improved upper and lower bounds on the determining number of Kneser graphs. Moreover, we provide the exact value of the determining number for some subfamilies of Kneser graphs.

math.CO↗