相关论文: Congruences for hook lengths of partitions

Recently Lachterman, Schayer, and Younger published an elegant proof of the Ramanujan congruences for the partition function $p(n)$. Their proof uses only the classical theory of modular forms as well as a beautiful result of Choie, Kohnen,…

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 $n$…

Simultaneous core partitions have attracted much attention since Anderson's work on the number of $(t_1,t_2)$-core partitions. In this paper we focus on simultaneous core partitions with distinct parts. The generating function of $t$-core…

Motivated by recent work of Hirschhorn and the author, Thejitha and Fathima recently considered arithmetic properties satisfied by the function $a_5(n)$ which counts the number of integer partitions of weight $n$ wherein even parts come in…

In recent work, Amdeberhan and Merca considered the integer partition function $a(n)$ which counts the number of integer partitions of weight $n$ wherein even parts come in only one color (i.e., they are monochromatic), while the odd parts…

In this paper, we use a branch of polyhedral geometry, Ehrhart theory, to expand our combinatorial understanding of congruences for partition functions. Ehrhart theory allows us to give a new decomposition of partitions, which in turn…

Previous work showed that, for $\nu_2(n)$ the number of partitions of $n$ into exactly two part sizes, one has $\nu_2(16n + 14) \equiv 0 \pmod{4}$. The earlier proof required the technology of modular forms, and a combinatorial proof was…

Motivated by Amdeberhan's conjecture on $(t,t+1)$-core partitions with distinct parts, various results on the numbers, the largest sizes and the average sizes of simultaneous core partitions with distinct parts were obtained by many…

The study of integer partitions and their congruences dates back to 1919 when Ramanujan discovered his famous congruences for the partition function, $p(n)$. Since then, many other kinds of partition functions have been discovered, as well…

Recently, the first author generalized a formula of Nekrasov and Okounkov which gives a combinatorial formula, in terms of hook lengths of partitions, for the coefficients of certain power series. In the course of this investigation, he…

In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called \emph{partitions with designated summands}. These are built by taking unrestricted integer partitions and designating exactly one of each occurrence…

Concave compositions are ordered partitions whose parts are decreasing towards a central part. We study the distribution modulo $a$ of the number of concave compositions. Let $c(n)$ be the number of concave compositions of $n$ having even…

Recently, Andrews and El Bachraoui considered the number of integer partitions whose smallest part is repeated exactly $k$ times and the remaining parts are not repeated. They presented several interesting results and posed questions…

Recently, using modular forms and Smoot's {\tt Mathematica} implementation of Radu's algorithm for proving partition congruences, Merca proved the following two congruences: For all $n\geq 0,$ \begin{align*} A(9n+5) & \equiv 0 \pmod{3}, \\…

In this paper, we count the total number of hooks of length two in all odd partitions of $n$ and all distinct partitions of $n$ with a bound on the largest part of the partitions. We generalize inequalities of Ballantine, Burson, Craig,…

The values of the partition function, and more generally the Fourier coefficients of many modular forms, are known to satisfy certain congruences. Results given by Ahlgren and Ono for the partition function and by Treneer for more general…

In this paper, we investigate the arithmetic properties of the difference between the number of partitions of a positive integer $n$ with even crank and those with odd crank, denoted $C(n)=c_e(n)-c_o(n)$. Inspired by Ramanujan's classical…

Noting a curious link between Andrews' even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is…

In 2015 Choi, Kim, and Lovejoy studied a weighted partition function, $A_1(m)$, which counted subpartitions with a structure related to the Rogers--Ramanujan identities. They conjectured the existence of an infinite class of congruences for…

We exactly determine the maximum possible hook length of $(s,t)$-core partitions with $d$-distinct parts when there are finitely many such partitions. Moreover, we provide an algorithm to construct a $d$-distinct $(s,t)$-core partition with…