The Inequalities of Merris and Foregger for Permanents

Conjectures on permanents are well-known unsettled conjectures in linear algebra. Let A be an n×n matrix and Sn be the symmetric group on n element set. The permanent of A is defined as perA=∑σ∈Sn∏i=1naiσ(i). The Merris conjectured that for all n×n doubly stochastic matrices (denoted by Ωn), nperA≥min1≤i≤n∑j=1nperA(j|i), where A(j|i) denotes the matrix obtained from A by deleting the jth row and ith column. Foregger raised a question whether per(tJn+(1−t)A)≤perA for 0≤t≤nn−1 and for all A∈Ωn, where Jn is a doubly stochastic matrix with each entry 1n. The Merris conjecture is one of the well-known conjectures on permanents. This conjecture is still open for n≥4. In this paper, we prove the Merris inequality for some classes of matrices. We use the sub permanent inequalities to prove our results. Foregger’s inequality is also one of the well-known inequalities on permanents, and it is not yet proved for n≥5. Using the concepts of elementary symmetric function and subpermanents, we prove the Foregger’s inequality for n=5 in [0.25, 0.6248]. Let σk(A) be the sum of all subpermanents of order k. Holens and Dokovic proposed a conjecture (Holen–Dokovic conjecture), which states that if A∈Ωn,A≠Jn and k is an integer, 1≤k≤n, then σk(A)≥(n−k+1)2nkσk−1(A). In this paper, we disprove the conjecture for n=k=4.

Two special cases of the Rational Shuffle Conjecture

International audience The Classical Shuffle Conjecture of Haglund et al. (2005) has a symmetric function side and a combinatorial side. The combinatorial side $q,t$-enumerates parking functions in the $n ×n$ lattice. The symmetric function side may be simply expressed as $∇ e_n$ , where $∇$ is the Macdonald eigen-operator introduced by Bergeron and Garsia (1999) and $e_n$ is the elementary symmetric function. The combinatorial side has been extended to parking functions in the $m ×n$ lattice for coprime $m,n$ by Hikita (2012). Recently, Gorsky and Negut have been able to extend the Shuffle Conjecture by combining their work (2012a, 2012b, 2013) (related to work of Schiffmann and Vasserot (2011, 2013)) with Hikita's combinatorial results. We prove this new conjecture for the cases $m=2$ and $n=2$ .

Counting strings over $\mathbb{Z}2^d$ with Given Elementary Symmetric Function Evaluations

International audience Let $\alpha$ be a string over $\mathbb{Z}_q$, where $q = 2^d$. The $j$-th elementary symmetric function evaluated at $\alpha$ is denoted $e_j(\alpha)$ . We study the cardinalities $S_q(m;\mathcal{T} _1,\mathcal{T} _2,\ldots,\mathcal{T} _t)$ of the set of length $m$ strings for which $e_j(\alpha) = \tau _i$. The $\textit{profile}$ k$(\alpha) = ⟨k_1,k_2,\ldots,k_(q-1) ⟩$ of a string $\alpha$ is the sequence of frequencies with which each letter occurs. The profile of $\alpha$ determines $e_j(\alpha)$ , and hence $S_q$. Let $h_n$ : $\mathbb{Z}_{2^{n+d-1}}^{(q-1)}$ $\mapsto \mathbb{Z}_{2^d} [z] $ mod $ z^{2^n}$ be the map that takes k$(\alpha)$ mod $2^{n+d-1}$ to the polynomial $1+ e_1(\alpha) z + e_2(\alpha) z^2 + ⋯+ e_{2^n-1}(\alpha)$ $z^{2^{n-1}}$. We show that $h_n$ is a group homomorphism and establish necessary conditions for membership in the kernel for fixed $d$. The kernel is determined for $d$ = 2,3. The range of $h_n$ is described for $d$ = 2. These results are used to efficiently compute $S_4(m;\mathcal{T} _1,\mathcal{T} _2,\ldots,\mathcal{T} _t)$ for $d$ = 2 and the number of complete factorizations of certain polynomials.

A Reverse Analytic Inequality for the Elementary Symmetric Function with Applications

We give a reverse inequality involving the elementary symmetric function by use of the Schur harmonic convexity theory. As applications, several new analytic inequalities for then-dimensional simplex are established.

Lyndon Words and Transition Matrices between Elementary, Homogeneous and Monomial Symmetric Functions

Let $h_\lambda$, $e_\lambda$, and $m_\lambda$ denote the homogeneous symmetric function, the elementary symmetric function and the monomial symmetric function associated with the partition $\lambda$ respectively. We give combinatorial interpretations for the coefficients that arise in expanding $m_\lambda$ in terms of homogeneous symmetric functions and the elementary symmetric functions. Such coefficients are interpreted in terms of certain classes of bi-brick permutations. The theory of Lyndon words is shown to play an important role in our interpretations.