相关论文: Elementary symmetric polynomials and a potentially…

Let e_k(x_1,...,x_l) be an elementary symmetric polynomial and let mu = (mu_1,...,mu_l) be an integer partition. Define pre_k(mu) to be the partition whose parts are the summands in the evaluation e_k(mu_1,...,mu_l). The study of such…

The elementary symmetric partition function is a map on the set of partitions. It sends a partition lambda to the partition whose parts are the summands in the evaluation of the elementary symmetric function on the parts of lambda. These…

Given a partition $\lambda$, we write $e_j(\lambda)$ for the $j^{\textrm{th}}$ elementary symmetric polynomial $e_j$ evaluated at the parts of $\lambda$ and $e_jp_A(n)$ for the sum of $e_j(\lambda)$ as $\lambda$ ranges over the set of…

Ballantine, Beck, and Merca defined a map $\mathrm{pre}_2$, which sends an integer partition $\lambda = (\lambda_1, \dots, \lambda_{\ell})$ to the set $\{\lambda_i\lambda_j : 1 \leq i < j \leq \ell\}$ consisting of the pairwise products of…

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,…

Introduced by Goulden and Jackson in their 1996 paper, the matchings-Jack conjecture and the hypermap-Jack conjecture (also known as the $b$-conjecture) are two major open questions relating Jack symmetric functions, the representation…

It is known that the elementary symmetric polynomials $e_k(x)$ have the property that if $ x, y \in [0,\infty)^n$ and $e_k(x) \leq e_k(y)$ for all $k$, then $||x||_p \leq ||y||_p$ for all real $0\leq p \leq 1$, and moreover $||x||_p \geq…

In this paper we obtained an original integer sequence based on the properties of the multinomial coefficient. We investigated a property of the sequence that shows connection with a primality testing. For any prime n the n-th term in the…

We consider two important families of BC_n-symmetric polynomials, namely Okounkov's interpolation polynomials and Koornwinder's orthogonal polynomials. We give a family of difference equations satisfied by the former, as well as…

We give combinatorial proofs of two identities from the representation theory of the partition algebra $C A_k(n), n \ge 2k$. The first is $n^k = \sum_\lambda f^\lambda m_k^\lambda$, where the sum is over partitions $\lambda$ of $n$,…

The pentagram map is a natural iteration on projective equivalence classes of (twisted) n-gons in the projective plane. It was recently proved ([OST]) that the pentagram map is completely integrable, with the complete set of Poisson…

In [7], Higashitani, Kummer, and Micha{\l}ek pose a conjecture about the symmetric edge polytopes of complete multipartite graphs and confirm it for a number of families in the bipartite case. We confirm that conjecture for a number of new…

In this note, we give a simple extension map from partitions of subsets of [n] to partitions of [n+1], which sends $\delta$-distant k-crossings to $(\delta+1)$-distant k-crossings (and similarly for nestings). This map provides a…

Let $\Lambda$ be the space of symmetric functions and $V_k$ be the subspace spanned by the modified Schur functions $\{S_\lambda[X/(1-t)]\}_{\lambda_1\leq k}$. We introduce a new family of symmetric polynomials,…

Ballantine--Beck--Feigon--Maurischat introduced the subsum polynomial \[ \operatorname{sp}(\lambda,x):=\prod_i (1+x^{\lambda_i}) \] attached to an integer partition $\lambda$, and studied rational functions obtained by summing reciprocals…

We study the factorization of Schubert polynomials into elementary symmetric polynomials. We conjecture that this occurs when the permutation corresponding to the Schubert polynomial does not contain the patterns $1432$, $1423$, $4132$, and…

One of our result is that 5 measurable sets in $R^8$ always admit an equipartition by 2 hyperplanes. This is an instance of a general equipartition problem (formulated by B. Gr{\" u}nbaum and H. Hadwiger) which can be reduced to the…

We consider a dominance order on positive vectors induced by the elementary symmetric polynomials. Under this dominance order we provide conditions that yield simple proofs of several monotonicity questions. Notably, our approach yields a…

Let $p_n$ be the number of partitions of an integer $n$. For each of the partition statistics of counting their parts, ranks, or cranks, there is a natural family of integer polynomials. We investigate their asymptotics and the limiting…

We utilize Dyson's concept of the adjoint of a partition to derive an infinite family of new polynomial analogues of Euler's Pentagonal Number Theorem. We streamline Dyson's bijection relating partitions with crank <= k and those with k in…