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Register a new userA Unification of Two Refinements of Eulers Partition Theorem
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We obtain a unification of two refinements of Eulers partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodts insertion algorithm for a generalization of the Andrews-Olsson partition identity is used in our combinatorial construction.
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In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider various sets of weighted tilings of a $1 times infty$ board with squares and dominoes, and for each type of tiling they construct a generating function in two different ways, which generates a $q$-series identity. Using this method, they recover quite a few classical $q$-series identities, but Eulers Pentagonal Number Theorem is not among them. In this paper, we introduce a key parameter when constructing the generating functions of various sets of tilings which allows us to recover Eulers Pentagonal Number Theorem along with an infinite family of generalizations.
For each positive integer $n$, we construct a bijection between the odd partitions and the distinct partitions of $n$ which extends Bressouds bijection between the odd-and-distinct partitions of $n$ and the splitting partitions of $n$. We compare our bijection with the classical bijections of Glaisher and Sylvester, and also with a recent bijection due to Chen, Gao, Ji and Li.
In this paper, we study growth rate of product of sets in the Heisenberg group over finite fields and the complex numbers. More precisely, we will give improvements and extensions of recent results due to Hegyv{a}ri and Hennecart (2018).
We prove a version of the Cauchy-Davenport theorem for general linear maps. For subsets $A,B$ of the finite field $mathbb{F}_p$, the classical Cauchy-Davenport theorem gives a lower bound for the size of the sumset $A+B$ in terms of the sizes of the sets $A$ and $B$. Our theorem considers a general linear map $L: mathbb{F}_p^n to mathbb{F}_p^m$, and subsets $A_1, ldots, A_n subseteq mathbb{F}_p$, and gives a lower bound on the size of $L(A_1 times A_2 times ldots times A_n)$ in terms of the sizes of the sets $A_1, ldots, A_n$. Our proof uses Alons Combinatorial Nullstellensatz and a variation of the polynomial method.
The Tur{a}n inequalities and the higher order Tur{a}n inequalities arise in the study of Maclaurin coefficients of an entire function in the Laguerre-P{o}lya class. A real sequence ${a_{n}}$ is said to satisfy the Tur{a}n inequalities if for $ngeq 1$, $a_n^2-a_{n-1}a_{n+1}geq 0$. It is said to satisfy the higher order Tur{a}n inequalities if for $ngeq 1$, $4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2geq 0$. A sequence satisfying the Turan inequalities is also called log-concave. For the partition function $p(n)$, DeSalvo and Pak showed that for $n>25$, the sequence ${ p(n)}_{n> 25}$ is log-concave, that is, $p(n)^2-p(n-1)p(n+1)>0$ for $n> 25$. It was conjectured by Chen that $p(n)$ satisfies the higher order Tur{a}n inequalities for $ngeq 95$. In this paper, we prove this conjecture by using the Hardy-Ramanujan-Rademacher formula to derive an upper bound and a lower bound for $p(n+1)p(n-1)/p(n)^2$. Consequently, for $ngeq 95$, the Jensen polynomials $g_{3,n-1}(x)=p(n-1)+3p(n)x+3p(n+1)x^2+p(n+2)x^3$ have only real zeros. We conjecture that for any positive integer $mgeq 4$ there exists an integer $N(m)$ such that for $ngeq N(m) $, the polynomials $sum_{k=0}^m {mchoose k}p(n+k)x^k$ have only real zeros. This conjecture was independently posed by Ono.
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