Related papers: Partition Statistics Equidistributed with the Numb…

Let $A_g$ be an abelian variety of dimension $g$ and $p$-rank $\lambda \leq 1$ over an algebraically closed field of characteristic $p>0$. We compute the number of homomorphisms from $\pi_1^{\text{\'et}}(A_g,a)$ to $GL_n(\mathbb F_q)$,…

An $L(h_1, h_2, \ldots, h_l)$-labelling of a graph $G$ is a mapping $\phi: V(G) \rightarrow \{0, 1, 2, \ldots\}$ such that for $1\le i\le l$ and each pair of vertices $u, v$ of $G$ at distance $i$, we have $|\phi(u) - \phi(v)| \geq h_i$.…

The $k$-measure of an integer partition was recently introduced by Andrews, Bhattacharjee and Dastidar. In this paper, we establish trivariate generating function identities counting both the length and the $k$-measure for partitions and…

Franklin's identity generalizes Euler's identity and states that the number of partitions of $n$ with $j$ different parts divisible by $r$ equals the number of partitions of $n$ with $j$ repeated parts. In this article, we give a refinement…

We prove polynomial boson-fermion identities for the generating function of the number of partitions of $n$ of the form $n=\sum_{j=1}^{L-1} j f_j$, with $f_1\leq i-1$, $f_{L-1} \leq i'-1$ and $f_j+f_{j+1}\leq k$. The bosonic side of the…

Under general conditions, the equation $g(x,y) = 0$ implicitly defines $y$ locally as a function of $x$. In this short note we study the combinatorial structure underlying a recently discovered formula for the divided differences of $y$…

Ferrers graphs and tables of partitions are treated as vectors. Matrix operations are used for simple proofs of identities concerning partitions. Interpreting partitions as vectors gives a possibility to generalize partitions on negative…

Whenever graphs admit equitable partitions, their quotient graphs highlight the structure evidenced by the partition. It is therefore very natural to ask what can be said about two graphs that have the same quotient according to certain…

Recently Corteel and Welsh outlined a technique for finding new sum-product identities by using functional relations between generating functions for cylindric partitions and a theorem of Borodin. Here, we extend this framework to include…

The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. It contains as subalgebras a large class of diagram algebras including the Brauer, planar partition,…

Let $t\geq2$ and $k\geq1$ be integers. A $t$-regular partition of a positive integer $n$ is a partition of $n$ such that none of its parts is divisible by $t$. Let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regular…

The \emph{$q,t$-Catalan numbers} $C_n(q,t)$ are polynomials in $q$ and $t$ that reduce to the ordinary Catalan numbers when $q=t=1$. These polynomials have important connections to representation theory, algebraic geometry, and symmetric…

Let $G_1, \dots, G_k$ be vector spaces over a finite field $\mathbb{F} = \mathbb{F}_q$ with a non-trivial additive character $\chi$. The analytic rank of a multilinear form $\alpha \colon G_1 \times \dots \times G_k \to \mathbb{F}$ is…

This paper shows that the number of hooks of length k contained in all partitions of n equals k times the number of parts of length k in all partitions of n. It contains also formulas for the moments (under uniform distribution) of k-th…

If $\gcd(r,t)=1$, then a theorem of Alladi offers the M\"obius sum identity $$-\sum_{\substack{ n \geq 2 \\ p_{\rm{min}}(n) \equiv r \pmod{t}}} \mu(n)n^{-1}= \frac{1}{\varphi(t)}. $$ Here $p_{\rm{min}}(n)$ is the smallest prime divisor of…

Let $\mathcal{B}(n)$ denote the collection of all set partitions of $[n]$. Suppose $\mathcal{A} \subseteq \mathcal{B}(n)$ is a non-trivial $t$-intersecting family of set partitions i.e. any two members of $\A$ have at least $t$ blocks in…

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,…

Let $\Lambda$ be the set of partitions of length $\geq 0$. We introduce an $\mathbb{N}$-graded algebra $\mathbb{A}_q^d(\Lambda)$ associated to $\Lambda$, which can be viewed as a quantization of the algebra of partitions defined by Reineke.…

We present an efficient sampling method for computing a partition function and accelerating configuration sampling. The method performs a random walk in the $\lambda$ space, with $\lambda$ being any thermodynamic variable that characterizes…

The restricted partitions in which the largest part is less than or equal to $N$ and the number of parts is less than or equal to $k$ were investigated by Andrews in \cite{Andrews76}. These partitions were extended recently by the author to…