Let $E_n^r=\{[τ]_a=(τ_1^{(a_1)},...,τ_n^{(a_n)})| τ\in S_n,\ 1\leq a_i\leq r\}$ be the set of all signed permutations on the symbols 1,2,...,n with signs 1,2,...,r. We prove, for every 2-letter signed pattern $[τ]_a$, that the number of $[τ]_a$-avoiding signed permutations in $E_n^r$ is given by the formula $\sum\limits_{j=0}^n j!(r-1)^j{n\choose j}^2$. Also we prove that there are only one Wilf class for r=1, four Wilf classes for r=2, and six Wilf classes for $r\geq 3$.
Signed Counting on Restricted Partitions and Combinatorial Proofs of Three Identities
College of Mathematics and Statistics & Center for Discrete Mathematics, Chongqing University（重庆大学数学与统计学院与离散数学中心）；College of Mathematics and Statistics, Chongqing University（重庆大学数学与统计学院）
In a graph $G$, a vertex dominates itself and its neighbors. A subset $D \subseteq V(G)$ is a double dominating set of $G$ if $D$ dominates every vertex of $G$ at least twice. A signed graph $Σ= (G,σ)$ is a graph $G$ together with an assignment $σ$ of positive or negative signs to all its edges. A cycle in a signed graph is positive if the product of its edge signs is positive. A signed graph is balanced if all its cycles are positive. A subset $D \subseteq V(Σ)$ is a double dominating set of $Σ$ if it satisfies the following conditions: (i) $D$ is a double dominating set of $G$, and (ii) $Σ[D…
Graph coverings are known to induce surjections of their critical groups. Here we describe the kernels of these morphisms in terms of data parametrizing the covering. Regular coverings are parametrized by voltage graphs, and the above kernel can be identified with a naturally defined voltage graph critical group. For double covers, the voltage graph is a signed graph, and the theory takes a particularly pleasant form, leading also to a theory of double covers of signed graphs.
We define a notion of vexillary signed permutation in types B, C, and D, corresponding to natural degeneracy loci for vector bundles with symmetries of those types. We show that the classes of these loci are given by explicit Pfaffian formulas. The Grassmannian formulas of Kazarian are important special cases, and the corresponding double Schubert polynomials of Ikeda, Mihalcea, and Naruse are shown to be equal to these Pfaffians.
We define a new statistic $\mathsf{sor}$ on the set of colored permutations $\mathsf{G}_{r,n}$ and prove that it has the same distribution as the length function. For the set of restricted colored permutations corresponding to the arrangements of $n$ non-attacking rooks on a fixed Ferrers shape we show that the following two sequences of set-valued statistics are joint equidistributed: $(\ell,\mathsf{Rmil}^0,\mathsf{Rmil}^1,...,\mathsf{Rmil}^{r-1}$, $\mathsf{Lmil}^0,\mathsf{Lmil}^1,...,\mathsf{Lmil}^{r-1}$, $\mathsf{Lmal}^0,\mathsf{Lmal}^1,...,\mathsf{Lmal}^{r-1}$, $\mathsf{Lmap}^0,\mathsf{Lma…
A shortest circuit cover ${\cal F}$ of a bridgeless graph $G$ is a family of circuits that covers every edge of $G$ and is of minimum total length. The total length of a shortest circuit cover ${\cal F}$ of $G$ is denoted by $SCC(G)$. For ordinary graphs (graphs without sign), the subject of shortest circuit cover is closely related to some mainstream areas, such as, Tutte's integer flow theory, circuit double cover conjecture, Fulkerson conjecture, and others. For signed graphs $G$, it is proved recently by Máčajová, Raspaud, Rollová and Škoviera that $SCC(G) \leq 11|E|$ if $G$ is s-bridgeles…
Previous work has studied the pattern count on singly restricted permutations. In this work, we focus on patterns of length 3 in multiply restricted permutations, especially for double and triple pattern-avoiding permutations. We derive explicit formulae or generating functions for various occurrences of length 3 patterns on multiply restricted permutations, as well as some combinatorial interpretations for non-trivial pattern relationships.
Recently, the study of patterns in inversion sequences was initiated by Corteel-Martinez-Savage-Weselcouch and Mansour-Shattuck independently. Motivated by their works and a double Eulerian equidistribution due to Foata (1977), we investigate several classical statistics on restricted inversion sequences that are either known or conjectured to be enumerated by {\em Catalan}, {\em Large Schröder}, {\em Baxter} and {\em Euler} numbers. One of the two highlights of our results is a fascinating bijection between $000$-avoiding inversion sequences and Simsun permutations, which together with Foata'…
For the calculation of Springer numbers (of root systems) of type $B_n$ and $D_n$, Arnold introduced a signed analogue of alternating permutations, called $β_n$-snakes, and derived recurrence relations for enumerating the $β_n$-snakes starting with $k$. The results are presented in the form of double triangular arrays ($v_{n,k}$) of integers, $1\le |k|\le n$. An Arnold family is a sequence of sets of such objects as $β_n$-snakes that are counted by $(v_{n,k})$. As a refinement of Arnold's result, we give analogous arrays of polynomials, defined by recurrence, for the calculation of the polynom…
A permutation $σ\in\mathfrak{S}_n$ is simsun if for all $k$, the subword of $σ$ restricted to $\{1,...,k\}$ does not have three consecutive decreasing elements. The permutation $σ$ is double simsun if both $σ$ and $σ^{-1}$ are simsun. In this paper we present a new bijection between simsun permutations and increasing 1-2 trees, and show a number of interesting consequences of this bijection in the enumeration of pattern-avoiding simsun and double simsun permutations. We also enumerate the double simsun permutations that avoid each pattern of length three.
We use Naito-Sagaki's work [S. Naito & D. Sagaki, J. Algebra 245 (2001) 395--412, J. Algebra 251 (2002) 461--474] on Lakshmibai-Seshadri paths fixed by diagram automorphisms to study the partitions fixed by Mullineux involution. We characterize the set of Mullineux-fixed partitions in terms of crystal graphs of basic representations of twisted affine Lie algebras of type $A_{2\ell}^{(2)}$ and of type $D_{\ell+1}^{(2)}$. We set up bijections between the set of symmetric partitions and the set of partitions into distinct parts. We propose a notion of double restricted strict partitions. Bijectio…
