Two permutations $\sigma,\tau\in S_n$ are called disjoint if the composition $\sigma\tau^{-1}$ has no fixed point. If a family $\mathcal F\subseteq S_n$ contains no $s$ pairwise disjoint permutations, then a simple averaging argument gives $|\mathcal F|\leq(s-1)(n-1)!$. Inozemtsev, Kolupaev and Kupavskii characterized the equality cases in the range $s\leq n/(2^{17}\log n).$ We characterize all equality cases throughout the range $2\le s\le n$: equality holds if and only if $\mathcal F$ is a union of $(s-1)$ pairwise disjoint $1$-cosets. We also prove the linear statement underlying this classification: a real-valued function on $S_n$ has constant sum on every one-factorization if and only if it lies in the span of the indicators of the $1$-cosets. The proof is combinatorial and applies to every order, with a few small orders handled separately.
Let $\mathrm{Sym(n,k)}$ denote the set of permutations on $\{1,2,\ldots,n\}$ with exactly $k$ cycles. A family $\mathcal{F}\subset\mathrm{Sym}(n,k)$ is said to be intersecting if $\sigma^{-1}\tau$ has a fixed point for all $\sigma,\tau\in\mathcal{F}$. In this paper, we investigate the size and structure of maximum-size...
Harwit and Sloane conjectured that every nonsingular entrywise-nonnegative matrix $A\in\mathbb R^{n\times n}$ satisfies $\|A^{-1}\|_F\ge 2n(n+1)^{-1}\|A\|_{\max}^{-1}$, with equality precisely for positive multiples of $S$-matrices. Cheng proved the conjecture in odd dimensions, while Frankel and Urschel proved the eve...
A family of permutations in $S_n$ is called even-intersecting if every two distinct members agree in an even number of positions. Let $M(n)$ denote the maximum size of such a family. For even $n$, we prove that $$n!!\leq M(n)\leq e^{\frac{n}{2}+o(n)}n!!,$$ improving the bound obtained from a theorem of Cameron, Deza an...
Anirban Banerjee, Abisek Dewan, R. Mishra· 0 citations
For a simple graph $G$ with $n$ vertices, write its chromatic polynomial in the rising factorial basis as $$ \chi_G(x)=\sum_{i=0}^{n}(-1)^{n-i}c_i(G)\langle x\rangle_i,$$ where $ \langle x\rangle_i=x(x+1)\cdots(x+i-1).$ The associated $\tau$-polynomial $$ \tau_G(x)=\sum_{i=0}^{n}c_i(G)x^i $$ was defined and systematica...
Ming-Yang Kang, Zhi-Xin Liu, Sophie C. C. Sun et al.· 0 citations
We show that for $n \geq 6$ each element of the commutator subgroup in the symmetric group $\mathfrak{S}_n$ resp. in the signed symmetric group $(\mathbb{Z}/2\mathbb{Z})^n\rtimes\mathfrak{S}_n$ is the commutator of two $n$-cycles resp. the commutator of two $n$-cycles with a negative sign product; with one exception. I...
P. Bader, Bernhard Böhmler, Patrick Wegener· 0 citations
For a positive integer $n$, let $f_n(X)=X^4-nX^3-6X^2+nX+1$ and let $K_n=\mathbb{Q}(\rho_n)$, where $\rho_n$ is a root of $f_n$. We determine all coincidences among these fields: for distinct positive integers $m,n$, $K_m=K_n \Longleftrightarrow \{m,n\}\in\{\{1,103\},\{2,22\},\{4,956\}\}$. Thus the three previously kno...
Zhilan Zhang· 0 citations
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