arXiv subjects

Manosij Ghosh Dastidar

Publications and source records attributed to Manosij Ghosh Dastidar.

12 recordsLinked to original sources

arXiv2026

The Partition Pairing Theorems I

The aim of this paper is to introduce pairing theory for partitions. We begin with two statistics on integer partitions, the \emph{pairing index} and the \emph{pairing rank}. The pairing index is equidistributed with the number of parts, while a joint refinement identifies its two constituents with the numbers of even and odd parts. We further introduce the \emph{pairing width} and prove that pairing index and pairing width are jointly equidistributed with the number of parts and the largest part. The resulting finite Gaussian generating function has a cyclotomic factorization from which Kummer's famous carry theorem for binomial coefficients follows. We also prove a mod-$5$ congruence for the excess of unpaired parts congruent to $1$ modulo $4$ over those congruent to $3$ modulo $4$ in the partitions of $5n+4$. A signed specialization exhibits that the parity of the pairing rank is governed by self-conjugate partitions. Motivated by this, we go on to introduce a second, diagrammatic pairing: after the two wings of the Durfee square are folded together, the unpaired cells break into connected \emph{diagonal blocks}. These blocks may be reflected independently, giving a Boolean decomposition of the set of partitions with a unique representative having all successive ranks nonnegative. We then relate our theory to overpartitions and Frobenius representations, obtaining as a corollary a geometric realization of overpartitions in terms of partitions whose principal hooks are all even. Finally, we study simply paired partitions of negative pairing rank, obtaining identities involving odd divisors and overpartitions, a parity theorem for pairing rank $-2$, and a Toeplitz determinant whose coefficientwise limit is an explicit infinite product related to MacMahon's product for plane partitions.

math.CO↗

arXiv2026

Sign Laws and Mock Theta Functions

Let \[ ρ(q)=\sum_{m\geq 0}\frac{q^{2m(m+1)}}{(1+q+q^2)(1+q^3+q^6)\cdots(1+q^{2m+1}+q^{4m+2})} =\sum_{n\geq 0}r(n)q^n \] be Ramanujan's third order mock theta function. We prove the sign law \[ r(3m)>0,\qquad r(3m+1)\leq 0,\qquad r(3m+2)\leq 0, \] with equality precisely at $n=2,4,8,11,20$. Watson's identity \[ 2ρ(q)+ω(q)=T(q) \] reduces the problem to comparing the mock theta function $ω(q)$ with the eta quotient \[ T(q)=3\frac{(q^6;q^6)_\infty^4}{(q^3;q^3)_\infty^2(q^2;q^2)_\infty}. \] We prove effective root-of-unity estimates for this difference. The polar contributions at $q=1$ cancel, the contribution at $q=-1$ is polynomially bounded, and the first surviving exponential term occurs at the primitive cubic roots of unity. It has the sign pattern \[ κ_0=\frac13\cos\fracπ{18}>0,\qquad κ_1=-\frac13\sin\frac{2π}{9}<0, \qquad κ_2=-\frac13\sin\frac\pi9<0. \] The resulting effective asymptotic proves the desired sign law for all sufficiently large $n$, and an exact integer-arithmetic verification completes the finite range. We conclude by indicating how the same root-of-unity method should lead to analogous sign laws for other third order mock theta functions, including $ϕ(q)$ and $χ(q)$.

math.NT↗

arXiv2026

Sign law for Ramanujan's third order mock theta function $ρ(q)$

We study the coefficients of Ramanujan's third order mock theta function \[ ρ(q)=\sum_{m\geq 0} \frac{q^{2m(m+1)}}{(1+q+q^2)(1+q^3+q^6)\cdots(1+q^{2m+1}+q^{4m+2})} =\sum_{n\geq 0}r(n)q^n. \] Numerical evidence suggests the striking sign pattern \[ r(3n)>0,\qquad r(3n+1)\leq 0,\qquad r(3n+2)\leq 0. \] We prove an asymptotic form of this phenomenon. More precisely, using Watson's relation between $ρ(q)$ and $ω(q)$, together with a Rademacher-type expansion for the coefficients of $ω(q)$ and the corresponding expansion for a theta--eta product, we show that \[ r(n)\sim κ_{n\bmod 3}\, \frac{2π}{(12n+8)^{1/4}} I_{1/2}\!\left(\frac{π\sqrt{12n+8}}{18}\right), \] where \[ κ_0=\frac13\cos\fracπ{18}>0, \qquad κ_1=-\frac13\sin\frac{2π}{9}<0, \qquad κ_2=-\frac13\sin\fracπ{9}<0. \] Consequently, \[ r(3n)>0, \qquad r(3n+1)<0, \qquad r(3n+2)<0 \] for all sufficiently large $n$.

math.NT↗

arXiv2024

Bijections between Variants of Dyck Paths and Integer Compositions

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 between these objects, such as congruence results.

math.CO↗

arXiv2024

Asymptotics of relaxed $k$-ary trees

A relaxed $k$-ary tree is an ordered directed acyclic graph with a unique source and sink in which every node has out-degree $k$. These objects arise in the compression of trees in which some repeated subtrees are factored and repeated appearances are replaced by pointers. We prove an asymptotic theta-result for the number of relaxed $k$-ary tree with $n$ nodes for $n \to \infty$. This generalizes the previously proved binary case to arbitrary finite arity, and shows that the seldom observed phenomenon of a stretched exponential term $e^{c n^{1/3}}$ appears in all these cases. We also derive the recurrences for compacted $k$-ary trees in which all subtrees are unique and minimal deterministic finite automata accepting a finite language over a finite alphabet.

math.CO↗

arXiv2024

Bijections and congruences involving lattice paths and integer compositions

We prove new bijections between different variants of Dyck paths and integer compositions, which give combinatorial explanations of their simple counting formula $4^{n-1}$. These give relations between different statistics, such as the number of crossings of the $x$-axis in classes of Dyck bridges or the distribution of peaks in classes of Dyck paths, and furthermore relate them with $k$- and $g$-compositions. These allow us to find and prove congruence results for Dyck paths and parity results for compositions. Our investigation uncovers unexpected connections to mock theta functions, Hardinian arrays, little Schröder paths, Fibonacci numbers, and irreducible pairs of compositions, offering new insights into the structures of paths, partitions and compositions.

math.CO↗

arXiv2022

Parity biases in partitions and restricted partitions

Let $p_{o}(n)$ (resp. $p_{e}(n)$) denote the number of partitions of $n$ with more odd parts (resp. even parts) than even parts (resp. odd parts). Recently, Kim, Kim, and Lovejoy proved that $p_{o}(n)>p_{e}(n)$ for all $n>2$ and conjectured that $d_{o}(n)>d_{e}(n)$ for all $n>19$ where $d_{o}(n)$ (resp. $d_{e}(n)$) denote the number of partitions into distinct parts having more odd parts (resp. even parts) than even parts (resp. odd parts). In this paper we provide combinatorial proofs for both the result and the conjecture of Kim, Kim and Lovejoy. In addition, we show that if we restrict the smallest part of the partition to be $2$, then the parity bias is reversed. That is, if $q_{o}(n)$ (resp. $q_{e}(n)$) denote the number of partitions of $n$ with more odd parts (resp. even parts) than even parts (resp. odd parts) where the smallest part is at least $2$, then we have $q_o(n) 7$. We also look at some more parity biases in partitions with restricted parts.

math.CO↗

arXiv2017

Some congruences modulo 5 and 25 for overpartition

We present two new Ramanujan-type congruences modulo 5 for overpartition. We also give an affirmative answer to a conjecture of Dou and Lin, which includes four congruences modulo 25 for overpartition.

math.NT↗

arXiv2017

Congruences and recursions for the cubic partition

Let $p_2(n)$ denote the number of cubic partitions. In this paper, we shall present two new congruences modulo $11$ for $p_2(n)$. We also provide an elementary alternative proof of a congruence established by Chan. Furthermore, we will establish a recursion for $p_2(n)$, which is a special case of a broader class of recursions.

math.NT↗

arXiv2016

Some results on the cubic partition

In this paper we explore Kruyswijk's method and show how to obtain congruences for cubic partition. That apart we also examine inequalities for a(n) and provide upper bound for it in the fashion of the classic partition function p(n).

math.NT↗

arXiv2011

Generalization of a few results in Integer Partitions

In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley's theorem and Elder's theorem, and the congruence results proposed by Ramanujan for the partition function. We generalize the results of Stanley and Elder from a fixed integer to an array of subsequent integers, and propose an analogue of Ramanujan's congruence relations for the `number of parts' function instead of the partition function. We also deduce the generating function for the `number of parts', and relate the technical results with their graphical interpretations through a novel use of the Ferrer's diagrams.

cs.DM↗

arXiv2010

Extension of Stanley's Theorem for Partitions

In this paper we present an extension of Stanley's theorem related to partitions of positive integers. Stanley's theorem states a relation between "the sum of the numbers of distinct members in the partitions of a positive integer $n$" and "the total number of 1's that occur in the partitions of $n$". Our generalization states a similar relation between "the sum of the numbers of distinct members in the partitions of $n$" and the total number of 2's or 3's or any general $k$ that occur in the partitions of $n$ and the subsequent integers. We also apply this result to obtain an array of interesting corollaries, including alternate proofs and analogues of some of the very well-known results in the theory of partitions. We extend Ramanujan's results on congruence behavior of the 'number of partition' function $p(n)$ to get analogous results for the 'number of occurrences of an element $k$ in partitions of $n$'. Moreover, we present an alternate proof of Ramanujan's results in this paper.

cs.DM↗