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 Annals of CombinatoricsJournal Prestige (SJR): 0.932 Citation Impact (citeScore): 1Number of Followers: 3      Hybrid journal (It can contain Open Access articles) ISSN (Print) 0219-3094 - ISSN (Online) 0218-0006 Published by Springer-Verlag  [2658 journals]
• On Reconstruction of Normal Edge-Transitive Cayley Graphs

Abstract: The main idea of this paper is to provide an algebraic algorithm for constructing symmetric graphs with optimal fault tolerance. For this purpose, we use normal edge-transitive Cayley graphs and the idea of reconstruction question posed by Praeger to present a special factorization of groups which induces a graphical decomposition of normal edge-transitive Cayley graphs to simpler normal edge-transitive Cayley graphs. Then as a consequence of our results, we continue the study of normal edge-transitive Cayley graphs of abelian groups and we show that knowing normal edge-transitive Cayley graphs of abelian p-groups, we can determine all normal edge-transitive Cayley graphs of abelian groups.
PubDate: 2020-10-12
DOI: 10.1007/s00026-020-00514-3

• Locks Fit into Keys: A Crystal Analysis of Lock Polynomials

Abstract: Lock polynomials and lock tableaux are natural analogues to key polynomials and Kohnert tableaux, respectively. In this paper, we compare lock polynomials to the much-studied key polynomials and give an explicit description of a crystal structure on lock tableaux. Furthermore, we construct an injective, weight-preserving map from lock tableaux to Kohnert tableaux that intertwines with their respective crystal operators. As a result, we see that the crystal structure on lock tableaux has a natural embedding into the Demazure crystal. We also examine the conditions for which key and lock polynomials are symmetric or quasisymmetric.
PubDate: 2020-10-12
DOI: 10.1007/s00026-020-00513-4

• Chain Decompositions of q ,  t -Catalan Numbers via Local Chains

Abstract: The q, t-Catalan number $${{\,\mathrm{Cat}\,}}_n(q,t)$$ enumerates integer partitions contained in an $$n\times n$$ triangle by their dinv and external area statistics. The paper by Lee et al. (SIAM J Discr Math 32:191–232, 2018) proposed a new approach to understanding the symmetry property $${{\,\mathrm{Cat}\,}}_n(q,t)={{\,\mathrm{Cat}\,}}_n(t,q)$$ based on decomposing the set of all integer partitions into infinite chains. Each such global chain $$\mathcal {C}_{\mu }$$ has an opposite chain $$\mathcal {C}_{\mu ^*}$$ ; these combine to give a new small slice of $${{\,\mathrm{Cat}\,}}_n(q,t)$$ that is symmetric in q and t. Here, we advance the agenda of Lee et al. (SIAM J Discr Math 32:191–232, 2018) by developing a new general method for building the global chains $$\mathcal {C}_{\mu }$$ from smaller elements called local chains. We define a local opposite property for local chains that implies the needed opposite property of the global chains. This local property is much easier to verify in specific cases compared to the corresponding global property. We apply this machinery to construct all global chains for partitions with deficit at most $$11$$ . This proves that for all n, the terms in $${{\,\mathrm{Cat}\,}}_n(q,t)$$ of degree at least $$\left( {\begin{array}{c}n\\ 2\end{array}}\right) -11$$ are symmetric in q and t.
PubDate: 2020-10-04
DOI: 10.1007/s00026-020-00512-5

• Lacunarity of Han–Nekrasov–Okounkov q -Series

Abstract: A power series is called lacunary if “almost all” of its coefficients are zero. Integer partitions have motivated the classification of lacunary specializations of Han’s extension of the Nekrasov–Okounkov formula. More precisely, we consider the modular forms \begin{aligned}F_{a,b,c}(z) :=\frac{\eta (24az)^a \eta (24acz)^{b-a}}{\eta (24z)},\end{aligned} defined in terms of the Dedekind $$\eta$$ -function, for integers $$a,c \ge 1$$ , where $$b \ge 1$$ is odd throughout. Serre (Publications Mathématiques de l’IHÉS 123–201:2959–2968, 1981) determined the lacunarity of the series when $$a = c = 1$$ . Later, Clader et al. (Am Math Soc 137(9):2959–2968, 2009) extended this result by allowing a to be general and completely classified the $$F_{a,b,1}(z)$$ which are lacunary. Here, we consider all c and show that for $${a \in \{1,2,3\}}$$ , there are infinite families of lacunary series. However, for $$a \ge 4$$ , we show that there are finitely many triples (a, b, c) such that $$F_{a,b,c}(z)$$ is lacunary. In particular, if $$a \ge 4$$ , $$b \ge 7$$ , and $$c \ge 2$$ , then $$F_{a,b,c}(z)$$ is not lacunary. Underlying this result is the proof the t-core partition conjecture proved by Granville and Ono (Trans Am Math Soc 348(1):331–347, 1996).
PubDate: 2020-09-24
DOI: 10.1007/s00026-020-00505-4

• Gamma Positivity of the Excedance-Based Eulerian Polynomial in Positive
Elements of Classical Weyl Groups

Abstract: The Eulerian polynomial $$\mathrm {AExc}_n(t)$$ enumerating excedances in the symmetric group $$\mathfrak {S}_n$$ is known to be gamma positive for all n. When enumeration is done over the type B and type D Coxeter groups, the type B and type D Eulerian polynomials are also gamma positive for all n. We consider $$\mathrm {AExc}_n^+(t)$$ and $$\mathrm {AExc}_n^-(t)$$ , the polynomials which enumerate excedance in the alternating group $$\mathcal {A}_n$$ and in $$\mathfrak {S}_n - \mathcal {A}_n$$ , respectively. We show that $$\mathrm {AExc}_n^+(t)$$ is gamma positive iff $$n \ge 5$$ is odd. When $$n \ge 4$$ is even, $$\mathrm {AExc}_n^+(t)$$ is not even palindromic, but we show that it is the sum of two gamma positive summands. An identical statement is true about $$\mathrm {AExc}_n^-(t)$$ . We extend similar results to the excedance based Eulerian polynomial when enumeration is done over the positive elements in both type B and type D Coxeter groups. Gamma positivity results are known when excedance is enumerated over derangements in $$\mathfrak {S}_n$$ . We extend some of these to the case when enumeration is done over even and odd derangements in $$\mathfrak {S}_n$$ .
PubDate: 2020-09-20
DOI: 10.1007/s00026-020-00511-6

• The Passing of Ron Graham

PubDate: 2020-09-07
DOI: 10.1007/s00026-020-00508-1

• Matching Numbers and the Regularity of the Rees Algebra of an Edge Ideal

Abstract: The regularity $${\text {reg}}R(I(G))$$ of the Rees ring R(I(G)) of the edge ideal I(G) of a finite simple graph G is studied. It is shown that, if R(I(G)) is normal, one has $${\text {mat}}(G) \le {\text {reg}}R(I(G)) \le {\text {mat}}(G) + 1$$ , where $${\text {mat}}(G)$$ is the matching number of G. In general, the induced matching number is a lower bound for the regularity, which can be shown by applying the squarefree divisor complex.
PubDate: 2020-09-07
DOI: 10.1007/s00026-020-00499-z

• F -Matrices of Cluster Algebras from Triangulated Surfaces

Abstract: For a given marked surface (S, M) and a fixed tagged triangulation T of (S, M), we show that each tagged triangulation $$T'$$ of (S, M) is uniquely determined by the intersection numbers of tagged arcs of T and tagged arcs of $$T'$$ . As a consequence, each cluster in the cluster algebra $${{\,\mathrm{{\mathcal {A}}}\,}}(T)$$ is uniquely determined by its F-matrix which is a new numerical invariant of the cluster introduced by Fujiwara and Gyoda.
PubDate: 2020-09-03
DOI: 10.1007/s00026-020-00507-2

• Balanced and Bruhat Graphs

Abstract: We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets having an R-labeling, imply the existence of the (non-homogeneous) $$\mathbf{c}\mathbf{d}$$ -index, a key invariant for studying inequalities for the flag vector of polytopes. Mirroring Alexander duality for Eulerian posets, we show an analogue of Alexander duality for bounded balanced digraphs. For Bruhat graphs of Coxeter groups, an important family of balanced graphs, our theory gives elementary proofs of the existence of the complete $$\mathbf{c}\mathbf{d}$$ -index and its properties. We also introduce the rising and falling quasisymmetric functions of a labeled acyclic digraph and show they are Hopf algebra homomorphisms mapping balanced digraphs to the Stembridge peak algebra. We conjecture non-negativity of the $$\mathbf{c}\mathbf{d}$$ -index for acyclic digraphs having a balanced linear edge labeling.
PubDate: 2020-08-29
DOI: 10.1007/s00026-020-00510-7

• Polynomization of the Bessenrodt–Ono Inequality

Abstract: In this paper, we investigate a generalization of the Bessenrodt–Ono inequality by following Gian–Carlo Rota’s advice in studying problems in combinatorics and number theory in terms of roots of polynomials. We consider the number of k-colored partitions of n as special values of polynomials $$P_n(x)$$ . We prove for all real numbers $$x >2$$ and $$a,b \in \mathbb {N}$$ with $$a+b >2$$ the inequality: \begin{aligned} P_a(x) \, \cdot \, P_b(x) > P_{a+b}(x). \end{aligned} We show that $$P_n(x) < P_{n+1}(x)$$ for $$x \ge 1$$ , which generalizes $$p(n) < p(n+1)$$ , where p(n) denotes the partition function. Finally, we observe for small values, the opposite can be true, since, for example: $$P_2(-3+ \sqrt{10}) = P_{3}(-3 + \sqrt{10})$$ .
PubDate: 2020-08-25
DOI: 10.1007/s00026-020-00509-0

• Cyclic Flats of a Polymatroid

Abstract: Polymatroids can be considered as “fractional matroids” where the rank function is not required to be integer valued. Many, but not every notion in matroid terminology translates naturally to polymatroids. Defining cyclic flats of a polymatroid carefully, the characterization by Bonin and de Mier of the ranked lattice of cyclic flats carries over to polymatroids. The main tool, which might be of independent interest, is a convolution-like method which creates a polymatroid from a ranked lattice and a discrete measure. Examples show the ease of using the convolution technique.
PubDate: 2020-08-12
DOI: 10.1007/s00026-020-00506-3

• A Noncommutative Cycle Index and New Bases of Quasi-symmetric Functions
and Noncommutative Symmetric Functions

Abstract: We define a new basis of the algebra of quasi-symmetric functions by lifting the cycle-index polynomials of symmetric groups to noncommutative polynomials with coefficients in the algebra of free quasi-symmetric functions, and then projecting the coefficients to QSym. By duality, we obtain a basis of noncommutative symmetric functions, for which a product formula and a recurrence in the form of a combinatorial complex are obtained. This basis allows to identify noncommutative symmetric functions with the quotient of $$\mathrm{FQSym}$$ induced by the pattern-replacement relation $$321\equiv 231$$ and $$312\equiv 132$$ .
PubDate: 2020-07-31
DOI: 10.1007/s00026-020-00504-5

• Combinatorial Interpretations of Lucas Analogues of Binomial Coefficients
and Catalan Numbers

Abstract: The Lucas sequence is a sequence of polynomials in s, t defined recursively by $$\{0\}=0$$ , $$\{1\}=1$$ , and $$\{n\}=s\{n-1\}+t\{n-2\}$$ for $$n\ge 2$$ . On specialization of s and t one can recover the Fibonacci numbers, the nonnegative integers, and the q-integers $$[n]_q$$ . Given a quantity which is expressed in terms of products and quotients of nonnegative integers, one obtains a Lucas analogue by replacing each factor of n in the expression with $$\{n\}$$ . It is then natural to ask if the resulting rational function is actually a polynomial in s, t with nonnegative integer coefficients and, if so, what it counts. The first simple combinatorial interpretation for this polynomial analogue of the binomial coefficients was given by Sagan and Savage, although their model resisted being used to prove identities for these Lucasnomials or extending their ideas to other combinatorial sequences. The purpose of this paper is to give a new, even more natural model for these Lucasnomials using lattice paths which can be used to prove various equalities as well as extending to Catalan numbers and their relatives, such as those for finite Coxeter groups.
PubDate: 2020-07-02
DOI: 10.1007/s00026-020-00500-9

• Ovals in $${\mathbb {Z}}^2_{2p}$$ Z 2 p 2

Abstract: By an oval in $${\mathbb {Z}}^2_{2p},$$ p odd prime, we mean a set of $$2p+2$$ points, such that no three of them are on a line. It is shown that ovals in $${\mathbb {Z}}^2_{2p}$$ only exist for $$p=3,5$$ and they are unique up to an isomorphism.
PubDate: 2020-07-02
DOI: 10.1007/s00026-020-00503-6

• On Inversion Triples and Braid Moves

Abstract: An inversion triple of an element w of a simply laced Coxeter group W is a set $$\{ \alpha , \beta , \alpha + \beta \}$$ , where each element is a positive root sent negative by w. We say that an inversion triple of w is contractible if there is a root sequence for w in which the roots of the triple appear consecutively. Such triples arise in the study of the commutation classes of reduced expressions of elements of W, and have been used to define or characterize certain classes of elements of W, e.g., the fully commutative elements and the freely braided elements. Also, the study of inversion triples is connected with the representation theory of affine Hecke algebras and double affine Hecke algebras. In this paper, we describe the inversion triples that are contractible, and we give a new, simple characterization of the groups W with the property that all inversion triples are contractible. We also study the natural action of W on the set of all triples of (not necessarily positive) roots of the form $$\{ \alpha , \beta , \alpha + \beta \}$$ . This enables us to prove rather quickly that every triple of positive roots $$\{ \alpha , \beta , \alpha + \beta \}$$ is contractible for some w in W and, moreover, when W is finite, w may be taken to be the longest element of W. At the end of the paper, we pose a problem concerning the aforementioned action.
PubDate: 2020-07-02
DOI: 10.1007/s00026-020-00501-8

• New Reduction Rules for the Tree Bisection and Reconnection Distance

Abstract: Recently it was shown that, if the subtree and chain reduction rules have been applied exhaustively to two unrooted phylogenetic trees, the reduced trees will have at most $$15k-9$$ taxa where k is the TBR (Tree Bisection and Reconnection) distance between the two trees, and that this bound is tight. Here, we propose five new reduction rules and show that these further reduce the bound to $$11k-9$$ . The new rules combine the “unrooted generator” approach introduced in Kelk and Linz (SIAM J Discrete Math 33(3):1556–1574, 2019) with a careful analysis of agreement forests to identify (i) situations when chains of length 3 can be further shortened without reducing the TBR distance, and (ii) situations when small subtrees can be identified whose deletion is guaranteed to reduce the TBR distance by 1. To the best of our knowledge these are the first reduction rules that strictly enhance the reductive power of the subtree and chain reduction rules.
PubDate: 2020-07-01
DOI: 10.1007/s00026-020-00502-7

• Kostant’s Partition Function and Magic Multiplex Juggling Sequences

Abstract: Kostant’s partition function is a vector partition function that counts the number of ways one can express a weight of a Lie algebra $$\mathfrak {g}$$ as a nonnegative integral linear combination of the positive roots of $$\mathfrak {g}$$ . Multiplex juggling sequences are generalizations of juggling sequences that specify an initial and terminal configuration of balls and allow for multiple balls at any particular discrete height. Magic multiplex juggling sequences generalize further to include magic balls, which cancel with standard balls when they meet at the same height. In this paper, we establish a combinatorial equivalence between positive roots of a Lie algebra and throws during a juggling sequence. This provides a juggling framework to calculate Kostant’s partition functions, and a partition function framework to compute the number of juggling sequences. From this equivalence we provide a broad range of consequences and applications connecting this work to polytopes, posets, positroids, and weight multiplicities.
PubDate: 2020-06-29
DOI: 10.1007/s00026-020-00498-0

• On Permutation Weights and q -Eulerian Polynomials

Abstract: Weights of permutations were originally introduced by Dugan et al. (Journal of Combinatorial Theory, Series A 164:24–49, 2019) in their study of the combinatorics of tiered trees. Given a permutation $$\sigma$$ viewed as a sequence of integers, computing the weight of $$\sigma$$ involves recursively counting descents of certain subpermutations of $$\sigma$$. Using this weight function, one can define a q-analog $$E_n(x,q)$$ of the Eulerian polynomials. We prove two main results regarding weights of permutations and the polynomials $$E_n(x,q)$$. First, we show that the coefficients of $$E_n(x, q)$$ stabilize as n goes to infinity, which was conjectured by Dugan et al. (Journal of Combinatorial Theory, Series A 164:24–49, 2019), and enables the definition of the formal power series $$W_d(t)$$, which has interesting combinatorial properties. Second, we derive a recurrence relation for $$E_n(x, q)$$, similar to the known recurrence for the classical Eulerian polynomials $$A_n(x)$$. Finally, we give a recursive formula for the numbers of certain integer partitions and, from this, conjecture a recursive formula for the stabilized coefficients mentioned above.
PubDate: 2020-06-06
DOI: 10.1007/s00026-020-00493-5

• New Moore-Like Bounds and Some Optimal Families of Abelian Cayley Mixed
Graphs

Abstract: Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are Cayley graphs of abelian groups. Such groups can be constructed using a generalization to $$\mathbb {Z}^n$$ of the concept of congruence in $$\mathbb {Z}$$. Here we use this approach to present some families of mixed graphs, which, for every fixed value of the degree, have an asymptotically large number of vertices as the diameter increases. In some cases, the results obtained are shown to be optimal.
PubDate: 2020-06-06
DOI: 10.1007/s00026-020-00496-2

• General Colored Partition Identities

Abstract: Ramanujan’s modular equations of degrees 3, 5, 7, 11 and 23 yield beautiful colored partition identities. Warnaar analytically generalized the modular equations of degrees 3 and 7,  and thereafter, the author found bijective proofs of those partitions identities and recently, established an analytic generalization of the modular equations of degrees 5, 11 and 23. The partition identities of degrees 5 and 11 were combinatorially proved by Sandon and Zanello, and it remains open to find a combinatorial proof of the partition identity of degree 23. In this paper, we prove general colored partition identities with a restriction on the number of parts, which are connected to the partition identities arising from those modular equations. We also provide bijective proofs of these partition identities. In particular, one of these proofs gives bijective proofs of the partition identity of degree 23 for some cases, which also work for the identities of degrees 5 and 11 for the same cases.
PubDate: 2020-06-03
DOI: 10.1007/s00026-020-00497-1

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