Related papers: Beck-type identities for Euler pairs of order $r$
The logical basis for information theory is the newly developed logic of partitions that is dual to the usual Boolean logic of subsets. The key concept is a "distinction" of a partition, an ordered pair of elements in distinct blocks of the…
A partition of $n$ is called a $t$-core partition if none of its hook number is divisible by $t.$ In 2019, Hirschhorn and Sellers \cite{Hirs2019} obtained a parity result for $3$-core partition function $a_3(n)$. Motivated by this result,…
We establish two general identities for Bernoulli and Euler polynomials, which are of a new type and have many consequences. The most striking result in this paper is as follows: If $n$ is a positive integer, $r+s+t=n$ and $x+y+z=1$, then…
Recently, Andrews and EI Bachraoui obtained several iden tities on two-colored partitions. While solving open problems they posed, Chen and Zhou derived a number of identities using analytic methods and asked for combinatorial proofs. In…
Recently, Merca and Yee proved some partition identities involving two new partition statistics. We refine these statistics and generalize the results of Merca and Yee. We also correct a small mistake in a result of Merca and Yee.
A partition is said to be $\ell$-regular if none of its parts is a multiple of $\ell$. Let $b^\prime_5(n)$ denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of $n$. This function has also close…
While there are many identities involving the Euler and Bernoulli numbers, they are usually proved analytically or inductively. We prove two identities involving Euler and Bernoulli numbers with combinatorial reasoning via up-down…
We show that, for a certain class of partitions and an even number of variables of which half are reciprocals of the other half, Schur polynomials can be factorized into products of odd and even orthogonal characters. We also obtain related…
In a general setting, we introduce a new bipartite state property sufficient for the validity of the perfect correlation form of the original Bell inequality for any three bounded quantum observables. A bipartite quantum state with this…
Kaur, Rana, and Eyyunni recently defined the mex sequence of a partition and established, by analytic methods, connections to two disparate types of partition-related objects. We make a bijection between partitions with certain mex…
For any integer $k\geq2$, we prove combinatorially the following Euler (binomial) transformation identity $$ \NC_{n+1}^{(k)}(t)=t\sum_{i=0}^n{n\choose i}\NW_{i}^{(k)}(t), $$ where $\NC_{m}^{(k)}(t)$ (resp.~$\NW_{m}^{(k)}(t)$) is the sum of…
In this paper a new method of experimental data analysis, the Particle-Set Identification method, is presented. The method allows to reconstruct moments of multiplicity distribution of identified particles. The difficulty the method copes…
When independent Bose-Einstein condensates (BEC), described quantum mechanically by Fock (number) states, are sent into interferometers, the measurement of the output port at which the particles are detected provides a binary measurement,…
A theorem of Andrews equates partitions in which no part is repeated more than 2k-1 times to partitions in which, if j appears at least k times, all parts less than j also do so. This paper proves the theorem bijectively, with some of the…
We prove new formulas for $p_d(n)$, the number of $d$-ary partitions of $n$, and, also, for its polynomial part. Given a partition $\lambda=(\lambda_1,\ldots,\lambda_{\ell})$, its associated $j$-th symmetric elementary partition,…
In a work of 1995, Alladi, Andrews, and Gordon provided a generalization of the two Capparelli identities involving certain classes of integer partitions. Inspired by that contribution, in particular as regards the general setting and the…
Motivated by recent study on the number of $t$-hooks in partitions arising from Euler's partition identity, we investigate the number of $t$-hooks in the sets from the first Rogers-Ramanujan identity and the first little G\"ollitz identity.…
We consider partitions of n-dimensional boxes in R^n, n>1, into a finite number of boxes with pairwise disjoint interiors. We study sets X \subseteq (0,\infty) with the Property (W_n): for every n-dimensional box P and every partition of P,…
Considering Schur positivity of differences of plethysms of homogeneous symmetric functions, we introduce a new relation on integer partitions. This relation is conjectured to be a partial order, with its restriction to one part partitions…
The Euler theorem in partition theory and its generalization are derived from a non-interacting quantum field theory in which each bosonic mode with a given frequency is equivalent to a sum of bosonic mode whose frequency is twice…