Related papers: A Method of Verifying Partition Congruences by Sym…
Categories of partitions are combinatorial structures arising from the representation theory of certain compact quantum groups and are linked to classical diagram algebras such as the Temperley-Lieb algebra. In this paper, we present…
This article introduces recursive relations allowing the calculation of the number of partitions with constraints on the minimum and/or on the maximum fragment size.
Recently, Pankaj Jyoti Mahanta and Manjil P. Saikika proved some identities relating certain restricted partitions into distinct odd parts with the partition whose odd parts are distinct combinatorially. They asked for the q-series proofs.…
A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number of new congruences akin to the classical Ramanujan-type. We…
Set partitions are arrangements of distinct objects into groups. The problem of listing all set partitions arises in a variety of settings, in particular in combinatorial optimization tasks. After a brief review, we give practical…
Designing quantum processors is a complex task that demands advanced verification methods to ensure their correct functionality. However, traditional methods of comprehensively verifying quantum devices, such as quantum process tomography,…
Quantum computers promise to efficiently solve not only problems believed to be intractable for classical computers, but also problems for which verifying the solution is also considered intractable. This raises the question of how one can…
We consolidate two widely believed conjectures about tautologies -- no optimal proof system exists, and most require superpolynomial size proofs in any system -- into a $p$-isomorphism-invariant condition satisfied by all paddable…
We show that, in many cases, there are infinitely many sets of partitions corresponding to a single analytical Rogers-Ramanujan type identity. This means that a single analytical Rogers-Ramanujan type identity implies the existence of…
We settle the so-called degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein {\it et al.} [Phys. Rev. A \textbf{73}, 012320 (2006)]. The conjecture states…
In this paper we present a novel algorithm for computing a congruence on an inverse semigroup from a collection of generating pairs. This algorithm uses a myriad of techniques from the theories of groups, automata, and inverse semigroups.…
Many complex networks show signs of modular structure, uncovered by community detection. Although many methods succeed in revealing various partitions, it remains difficult to detect at what scale some partition is significant. This problem…
Although the realization of useful quantum computers poses significant challenges, swift progress in emerging quantum technologies is making this goal realistically approachable. In this context, one of the essential resources is quantum…
In this paper we develop cyclic proof systems for the problem of inclusion between the least sets of models of mutually recursive predicates, when the ground constraints in the inductive definitions belong to the quantifier-free fragments…
New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.
In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or…
In this paper, we prove that almost every translation of $\mathbb{T}^2$ admits a symbolic coding which has linear complexity $2n+1$. The partitions are constructed with Rauzy fractals associated with sequences of substitutions, which are…
This paper deals with the algorithmic aspects of solving feasibility problems of semidefinite programming (SDP), aka linear matrix inequalities (LMI). Since in some SDP instances all feasible solutions have irrational entries, numerical…
This paper is concerned with certifying that a given point is near an exact root of an overdetermined or singular polynomial system with rational coefficients. The difficulty lies in the fact that consistency of overdetermined systems is…
A symbolic analysis of observed time series data requires making a discrete partition of a continuous state space containing observations of the dynamics. A particular kind of partition, called ``generating'', preserves all dynamical…