Related papers: Arithmetic of partitions and the $q$-bracket opera…
We study the semiclassical partition function in the frame work of the Morse theory, to clarify the phase factor of the partition function and to relate it to the eta invariant of Atiyah. Converting physical system with potential into a…
We examine "partition zeta functions" analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those…
We prove a multiplication theorem for quantum cluster algebras of acyclic quivers. The theorem generalizes the multiplication formula for quantum cluster variables in \cite{fanqin}. We apply the formula to construct some $\mathbb{ZP}$-bases…
In this note we give representations for the partition algebra A_3(Q) in Young's seminormal form. For this purpose, we also give characterizations of A_n(Q) and$A_{n-1/2}(Q).
The well-known difficulties arising in a classification which is not set-theoretically trivial---involving what is sometimes called a non-smooth quotient---have been overcome in a striking way in the theory of operator algebras by the use…
We establish a Pythagorean theorem for the absolute values of the blocks of a partitioned matrix. This leads to a series of remarkable operator inequalities.
We obtain partition regularity results for homogeneous quadratic equations whose parametrized solutions admit nice factorizations into linear forms over rings of integers of imaginary quadratic fields. To do so, we develop number-theoretic…
The famous partition theorem of Euler states that partitions of $n$ into distinct parts are equinumerous with partitions of $n$ into odd parts. Another famous partition theorem due to MacMahon states that the number of partitions of $n$…
We introduce an algebraic theory of integration on quantum planes and other braided spaces. In the one dimensional case we obtain a novel picture of the Jackson $q$-integral as indefinite integration on the braided group of functions in one…
We demonstrate that statistics for several types of set partitions are described by generating functions which appear in the theory of integrable equations.
Motivated by the sharp contrast between classical and quantum physics as probability theories, in these lecture notes I introduce the basic notions of operator algebras that are relevant for the algebraic approach to quantum physics.…
For $l,n \in \mathbb{N}$ we define tonal partition algebra $P^l_n$ over $\mathbb{Z}[\delta]$. We construct modules $\{ \Delta_{\underline{\mu}} \}_{\underline{\mu}}$ for $P^l_n$ over $\mathbb{Z}[\delta]$, and hence over any integral domain…
Numerical semigroups are cofinite additive submonoids of the natural numbers. In 2011, Keith and Nath illustrated an injection from numerical semigroups to integer partitions. We explore this connection between partitions and numerical…
We continue a series of papers studying the parity of families of eta-quotients, which provide implications for the parity of the partition function as well as an overarching conjecture on related $q$-series. The present article focuses on…
We identify a class of "semi-modular" forms invariant on special subgroups of $GL_2(\mathbb Z)$, which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an…
Although the study of functional calculus has already established necessary and sufficient conditions for operators to be fractionalized, this paper aims to use our well-conceived notion of integer powers of operators to construct…
We introduce, characterise and provide a combinatorial interpretation for the so-called $q$-Jacobi-Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order $q$-differential…
Using $P(n,m)$, the number of integer partitions of $n$ into exactly $m$ parts, which was the subject of an earlier paper, $P(n,m,p)$, the number of integer partitions of $n$ into exactly $m$ parts with each part at most $p$, can be…
By a re-examination of MacMahon's original proof of his celebrated theorem on the distribution of the major indices over permutations, we give a reformulation of his argument in terms of the structure of labeled partitions. In this…
In this paper we consider the $q$-Brauer algebra over $R$ a commutative noetherian domain. We first construct a new basis for $q$-Brauer algebras, and we then prove that it is a cell basis, and thus these algebras are cellular in the sense…