Let \(\mathcal A(2n)\) be the set of up--down alternating permutations\[\sigma_1<\sigma_2>\sigma_3<\sigma_4>\cdots<\sigma_{2n},\]and let\[E_{2n}(q)=\sum_{\sigma\in \mathcal A(2n)}q^{\operatorname{inv}(\sigma)}.\]We give a self-contained combinatorial proof of the congruence\[E_{2n}(q)\equiv q^{2n(n-1)}-\binom n2(1+q)^2\pmod{(1+q)^3}.\]This refines the Andrews-Foata congruence modulo \((1+q)^2\) for the \(q\)-secant numbers. The proof decomposes alternating permutations into orbits generated by switches of natural pairs \(\{2i-1,2i\}\). Three free natural pairs produce an orbit of type \((\mathbb Z/2\mathbb Z)^3\), and hence a factor \((1+q)^3\). The remaining contribution comes from permutations with exactly two free natural pairs; after forced outside blocks are removed, a sign-reversing sliding involution leaves one unpaired core, giving the correction term \(-\binom n2(1+q)^2\).
The families of graphs defined by a certain type of system of equations over commutative rings have been studied and used since the 1990s. This survey presents these families and their applications related to graphs, digraphs, and hypergraphs. Some open problems and conjectures are mentioned.
This paper establishes operad structures on the collection of poset matrices by introducing a new framework of partial composition operations. Extending the combinatorial setting of naturally labelled posets, we define several partial compositions that serve as basic tools for constructing poset matrices of arbitrary size. We prove that three of these operations satisfy the axioms of operads, thereby giving an operad structure to the set of poset matrices. We further characterize the dual operations and provide explicit combinatorial interpretations of these constructions in terms of naturally labelled posets. In addition, we address some open questions on poset enumeration via the operads of poset matrices.
The somewhere-to-below shuffles are the elements$$ t_{\ell} := \operatorname{cyc}_{\ell} + \operatorname{cyc}_{\ell,\ell+1} + \operatorname{cyc}_{\ell,\ell+1,\ell+2} + \cdots + \operatorname{cyc}_{\ell,\ell+1,\ldots,n} $$(for $\ell\in\left\{ 1,2,\ldots,n\right\} $) in the group algebra $\mathbf{k}\left[ S_{n}\right] $ of the $n$-th symmetric group $S_{n}$. Their linear combinations are called the \emph{one-sided cycle shuffles}. We determine the eigenvalues of the action of any one-sided cycle shuffle on any Specht module $\mathcal{S}^{\lambda}$ of $S_{n}$.
We introduce a linear extension of the rotor-routing model in directed graphs, akin to the sandpile model and vector addition systems, together with new rotor mechanisms that extend standard cyclic rotors. In this framework, rotor-routing is interpreted as the simultaneous movement of particles accross two coupled graphs, involving both vertex and arc-based particles. The standard, combinatorial rotorrouting of positive particles (legal routing) based on rotor-configurations, then becomes a special case of a linear equivalence. We give comprehensive reachability results characterizing legal routings among linear equivalences, expanding on previous results, and settle the algorithmic complexities associated with these problems.
We introduce phased multi de Bruijn sequences, a generalization of de Bruijn sequences. A phased string is a string whose positions sequentially rotate through several alphabets; e.g., "0Ax1Ay1By0Az1Bx" rotates through alphabets ohm 0 = {0, 1}, ohm 1 = {A, B}, and ohm 2 = {x, y, z}. We consider cyclic phased strings in which all possible phased k-mers (phased strings of length k) occur with particular multiplicities, depending on their "phase" (the alphabet they start in). For example, consider the cycle (s) =(0Ax1Ay1By0Az0By1Bx0Ay0Bx1Bz1Ax0Bz1Az) in these alphabets. All possible phased 2-mers starting in phases 0, 1, and 2 respectively have multiplicities 3, 2, and 2; e.g., "0A" occurs three times, "Ax" occurs twice, and "z0" occurs twice (including the occurrence that wraps around the cycle). We determine parameters (k, number of phases, alphabet sizes, and multiplicities) for which this is possible. Then we count the total number of phased multi de Bruijn sequences for these parameters. This extends classical de Bruijn sequences and multi de Bruijn sequences (our previous generalization of de Bruijn sequences in which all possible k-mers over one alphabet occur m times each). Our method of counting the sequences uses a change of basis for the Laplacian matrix; this also gives a new proof for the number of classical de Bruijn sequences, as they are a special case of this framework.
A LAnKe (also known as a Filippov algebra or a Lie algebra of the n-th kind) is a vector space equipped with a skew-symmetric n-linear form that satisfies the generalized Jacobi identity. Friedmann, Hanlon, Stanley and Wachs have shown that the symmetric group acts on the multilinear part of the free LAnKe on 2n - 1 generators as an irreducible representation. They announced that the multilinear component on 3n - 2 generators decomposes as a direct sum of two irreducible symmetric group representations and a proof was given recently in a subsequent paper by Friedmann, Hanlon and Wachs. In the present paper we provide a proof of the later statement. The two proofs are substantially different.
In the finite case, there are close connections between lattices and matroids. More notably, every lattice can be realised as the lattice of cyclic flats of some matroid. The main concern of this paper is finding analogous connections in the infinite case. In particular, we introduce a class of set systems (which forms a superclass of matroids) whose lattices of cyclic flats model all (possibly infinite) complete lattices.
It is known that the minimal volume entropy of a connected finite graph of a given cyclomatic number is attained by a trivalent graph endowed with its combinatorial length. The purpose of this short note is to present a simple geometric proof of this result based solely on elementary combinatorial arguments.
The polynomial reconstruction problem, proposed by Cvetkovic in 1973, asks whether the characteristic polynomial (G; x) of a graph G with at least 3 vertices can be reconstructed from the polynomial deck P(G) = { (G - vi; x)}vi is an element of V (G). In 2000, Hagos proposed a question of whether (G; x) is reconstructible from the Spier (2025) proved that the characteristic polynomials of two graphs are congruent modulo 4 if and only if the characteristic polynomials of their complements are congruent modulo 4. Motivated by the above, we prove that for any two graphs G and H, if (G; x)- (H; x) equivalent to c (mod 4) and ( G & strns;; x)- ( H & strns;; x) equivalent to d (mod 4) for two constants c and d, respectively, then c equivalent to d (mod 4). In particular, if n is even, then c equivalent to d equivalent to 0 (mod 4). This strengthens Spier's results, and provides some non-trivial information about the constant coefficients of (G; x) and (H; x) for any potential counterexample pair (G, H) to the polynomial reconstruction problem. We also obtain a similar result for any potential counterexample pair (G, H) to the problem proposed by Hagos in 2000.
Consider a finite field Fq, q = pd, where p is an odd prime. Let M = (E, r) be a regular matroid; denote by B the family of its bases, s & strns;(M; alpha) = where alpha e is an element of Fq, alpha e =/ 0. Let a subset A equivalent to A(alpha) in E have maximum cardinality and satisfy the condition s & strns;(M|A; alpha) =/ 0, while r & lowast;(alpha) = |A| - r(E). Let us represent the value of the characteristic polynomial of the matroid M at the point q as a linear combination of Legendre symbols with respect to s & strns;(M|A; alpha), whose coefficients are equal in modulus to 1/qr & lowast;(alpha)/2. This representation generalizes the formula for the flow polynomial of a graph which was obtained by us earlier. The latter formula is an analog of the so-called alpha-representation of vacuum Feynman amplitudes over finite fields, which inspired the Kontsevich conjecture (1997). The alpha-representation technique is also applicable to expressing the number of Tait colorings for a cubic biconnected planar graph in terms of principal minors of the face matrix of this graph.
Let G be a graph with adjacency eigenvalues lambda(1)>= & centerdot;& centerdot;& centerdot;>= lambda(n). Both lambda(1)+ lambda(n )and the odd girth of G can be seen as measures of the bipartiteness of G. Csikvari proved in 2022 that for odd girth 5 graphs (triangle-free) it holds that (lambda(1 )+ lambda(n))/n <= (3- 2 root 2) < 0.1716. In this paper we extend Csikvari's result to general odd girth kproving that (lambda(1)+lambda(n))/n = O(k(-1)). In the case of odd girth 7, we prove a stronger upper bound of (lambda(1)+ lambda(n))/n < 0.0396.
The class of fighting fish is a recently introduced model of branching surfaces constructed by gluing square cells in a directed way, that generalizes the standard combinatorial class of parallelogram polyominoes. Fighting fish are enumerated (3n with respect to their half-perimeter by the sequence 2 ), also known (n+1)(2n+1) n for counting non-separable rooted planar maps w.r.t the number of edges. Another common feature of these two combinatorial classes is that fighting fish can be seen as particular excursions on the square lattice confined to the quarter plane, while excursions on the quarter plane encode tree-rooted planar maps via Mullin's encoding. We build on this point of view and on recursive decompositions of fish and maps to show that fighting fish can be identified with the Lehman-Lenormand codes of non-separable rooted planar maps, which is obtained by endowing maps with their rightmost depth first search tree before applying Mullin's encoding. The resulting direct bijection yields a simple characterization of fighting fish as minimal non-separable excursions in the quarter plane. We then also introduce a natural extension of fighting fish that we call generalized fighting fish and we follow again the same approach to show that they correspond to Lehman-Lenormand codes of (general) rooted planar maps.
The type A Kostant partition function (KPF) enumerates several families of objects that arise in representation theory and combinatorics, including Tesler matrices, Kostant pictures, Lusztig data, and integral flows. In this paper, we establish logarithmic asymptotics for several classes of KPF by linking them to integer partitions. To this end we introduce height diagrams, integral and row Tesler matrices. As an application, we obtain the logarithmic asymptotics of regular Tesler matrices. We also compare the known poset structures on the aforementioned KPF objects. We prove that the Lusztig data partial order induced on Kostant pictures refines the natural partial order on those, which we also show to be equivalent to the partial order on Tesler matrices.
What is the least integer $\text{sd}(n)$ such that every graph on $n$ vertices has fractional chromatic number $p /q$, where $p$ and $q$ are positive integers and $q \le \text{sd}(n)$? An upper bound on the determinants of Hadamard matrices implies that $\text{sd}(n)\le 2^{-n}(n+1)^{(n+1) /2}$. The only known lower bound on $\text{sd}(n)$ that is exponential in $n$ (asymptotically, roughly $1.346^n/\sqrt{\log n}$) was obtained using an iterated Mycielski construction [D. C. Fisher, J. Graph Theory 20 (1995), 403-409]. We improve on this bound by constructing a family of graphs which shows that $\text{sd}(n) \geq 2^{n/2}$.
Given a tree T and a subtree S of T, one can define the local mean at S, & micro;T (S) , to be the average order of the subtrees of T containing S. In 1983, Jamison showed that & micro;(T) (S) < & micro;(T)(S') if S subset of S' as subtrees of T. Therefore, it is natural to ask the following question. Among all the k-subtrees (subtrees of order k), which one achieves the maximal/minimal local mean and what properties does it have? We call such k-subtrees k-maximal/k-minimal. Wagner and H. Wang showed in 2016 that a 1-maximal subtree has degree 1 or 2. In this paper, we show that if T is not a path, a 1-minimal subtree of T has degree at least 3. For k >= 2, we show that a k-maximal subtree has at most one leaf whose degree in T is greater than 2, and that such a leaf can only occur when all other leaves in S are also leaves in T. Parallel results hold for k-minimal subtrees. Roughly speaking, the leaves of a k-maximal subtree tend to have degree 1 or 2 in T, while the leaves of a k-minimal subtree tend to have degree at least 3 in T. In the second part, this paper introduces the local density as a normalization of local means, for the sake of comparing subtrees of different orders. We show that the local density at subtree S is lower-bounded by 1/2 with equality if and only if S contains all the vertices of degree at least 3 in T. On the other hand, local density can be arbitrarily close to 1.
Given a set S, an integer k and a permutation group G acting on S, we provide a novel algorithm for computing parts per thousand Sk & Zcaron;G, which is the set of all k-subsets of S up to symmetries in G. Then we apply our algorithm to compute (d, g)-cages which are d-regular graphs of girth g and minimum order n = n(d, g). Our algorithm allowed us to reproduce most of the computational results on cages and to improve six lower bounds for the order of cages, namely: n(3, 14) >= 262, n(3, 15) >= 388, n(3, 17) >= 770, n(4, 9) >= 165, n(5, 7) >= 110 and n(8, 5) >= 69.
We describe a relationship between the Lie algebra sl(4)(C) and the hypercube graphs. Consider the C-algebra P of polynomials in four commuting variables. We turn P into an sl(4)(C)-module on which each element of sl(4)(C) acts as a derivation. Then P becomes a direct sum of irreducible sl(4)(C)-modules P=& sum;P-N is an element of N(N), where PN is the Nth homogeneous component of P. For N is an element of N we construct some additional sl(4)(C)-modules Fix(G) and T. For these modules the underlying vector space is described as follows. Let X denote the vertex set of the hypercube H(N,2), and let V denote the C-vector space with basis X. For the automorphism group G of H(N,2), the action of G on X turns V into a G-module. The vector space V-circle times 3=V circle times V circle times V becomes a G-module such that g(u circle times v circle times w)=g(u)circle times g(v)circle times g(w) for g is an element of G and u,v,w is an element of V. The subspace Fix(G) of V circle times 3 consists of the vectors in V-circle times 3 that are fixed by every element in G. Pick kappa is an element of X. The corresponding subconstituent algebra T of H(N,2) is the subalgebra of End(V) generated by the adjacency map A of H(N,2) and the dual adjacency map A(& lowast;) of H(N,2) with respect to kappa. In our main results, we turn Fix(G) and T into sl(4)(C)-modules, and display sl(4)(C)-module isomorphisms P-N -> Fix(G)-> T. We describe the sl(4)(C)-modules PN, Fix(G), T from multiple points of view.
Let H = (H-i: i < alpha) for some ordinal number alpha be an indexed family of graphs. A family G = (G(i) : i < alpha) of edge-disjoint subgraphs of a graph G such that for every i < alpha: G(i) is isomorphic to H-i, each G(i) is a spanning subgraph of G, and E(G) = U{E(Gi) : i < alpha} is a H-factorization of G. Let kappa be an infinite cardinal. Konig proved in 1936 that every kappa-regular graph has a factorization into perfect matchings. We extend this result to the most general factorizations possible. We study indexed families T = (T-i: i < kappa) of graphs without isolated vertices such that every connected kappa-regular graph has a T-factorization. We prove that if T is a family of forests each of order at most kappa, then every connected kappa-regular graph G has a T-factorization. These are the most general assumptions for such a family T for this statement to hold.
