Annals of Combinatorics

Gallagher, Katherine; Li, Lucia; Vassilev, Katja

doi: 10.1007/s00026-020-00505-4pmid: N/A

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 Fa,b,c(z):=η(24az)aη(24acz)b-aη(24z),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned}F_{a,b,c}(z) :=\frac{\eta (24az)^a \eta (24acz)^{b-a}}{\eta (24z)},\end{aligned}$$\end{document}defined in terms of the Dedekind η\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\eta $$\end{document}-function, for integers a,c≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a,c \ge 1$$\end{document}, where b≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$b \ge 1$$\end{document} is odd throughout. Serre (Publications Mathématiques de l’IHÉS 123–201:2959–2968, 1981) determined the lacunarity of the series when a=c=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a = c = 1$$\end{document}. 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 Fa,b,1(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F_{a,b,1}(z)$$\end{document} which are lacunary. Here, we consider all c and show that for a∈{1,2,3}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${a \in \{1,2,3\}}$$\end{document}, there are infinite families of lacunary series. However, for a≥4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a \ge 4$$\end{document}, we show that there are finitely many triples (a, b, c) such that Fa,b,c(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F_{a,b,c}(z)$$\end{document} is lacunary. In particular, if a≥4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a \ge 4$$\end{document}, b≥7\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$b \ge 7$$\end{document}, and c≥2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$c \ge 2$$\end{document}, then Fa,b,c(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F_{a,b,c}(z)$$\end{document} 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).

Csirmaz, Laszlo

doi: 10.1007/s00026-020-00506-3pmid: N/A

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.

Gyoda, Yasuaki; Yurikusa, Toshiya

doi: 10.1007/s00026-020-00507-2pmid: N/A

For a given marked surface (S, M) and a fixed tagged triangulation T of (S, M), we show that each tagged triangulation T′\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$T'$$\end{document} of (S, M) is uniquely determined by the intersection numbers of tagged arcs of T and tagged arcs of T′\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$T'$$\end{document}. As a consequence, each cluster in the cluster algebra A(T)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\,\mathrm{{\mathcal {A}}}\,}}(T)$$\end{document} is uniquely determined by its F-matrix which is a new numerical invariant of the cluster introduced by Fujiwara and Gyoda.

Heim, Bernhard; Neuhauser, Markus; Tröger, Robert

doi: 10.1007/s00026-020-00509-0pmid: N/A

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 Pn(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P_n(x)$$\end{document}. We prove for all real numbers x>2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x >2 $$\end{document} and a,b∈N\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a,b \in \mathbb {N}$$\end{document} with a+b>2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a+b >2$$\end{document} the inequality: Pa(x)·Pb(x)>Pa+b(x).\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} P_a(x) \, \cdot \, P_b(x) > P_{a+b}(x). \end{aligned}$$\end{document}We show that Pn(x)<Pn+1(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P_n(x) < P_{n+1}(x)$$\end{document} for x≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x \ge 1$$\end{document}, which generalizes p(n)<p(n+1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p(n) < p(n+1)$$\end{document}, where p(n) denotes the partition function. Finally, we observe for small values, the opposite can be true, since, for example: P2(-3+10)=P3(-3+10)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P_2(-3+ \sqrt{10}) = P_{3}(-3 + \sqrt{10})$$\end{document}.

Dey, Hiranya Kishore; Sivasubramanian, Sivaramakrishnan

doi: 10.1007/s00026-020-00511-6pmid: N/A

The Eulerian polynomial AExcn(t)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \mathrm {AExc}_n(t)$$\end{document} enumerating excedances in the symmetric group Sn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathfrak {S}_n$$\end{document} 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 AExcn+(t)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \mathrm {AExc}_n^+(t)$$\end{document} and AExcn-(t)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \mathrm {AExc}_n^-(t)$$\end{document}, the polynomials which enumerate excedance in the alternating group An\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {A}_n$$\end{document} and in Sn-An\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathfrak {S}_n - \mathcal {A}_n$$\end{document}, respectively. We show that AExcn+(t)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \mathrm {AExc}_n^+(t)$$\end{document} is gamma positive iff n≥5\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n \ge 5$$\end{document} is odd. When n≥4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n \ge 4$$\end{document} is even, AExcn+(t)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \mathrm {AExc}_n^+(t)$$\end{document} is not even palindromic, but we show that it is the sum of two gamma positive summands. An identical statement is true about AExcn-(t)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \mathrm {AExc}_n^-(t)$$\end{document}. 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 Sn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathfrak {S}_n$$\end{document}. We extend some of these to the case when enumeration is done over even and odd derangements in Sn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathfrak {S}_n$$\end{document}.

Han, Seongjune; Lee, Kyungyong; Li, Li; Loehr, Nicholas A.

doi: 10.1007/s00026-020-00512-5pmid: N/A

The q, t-Catalan number Catn(q,t)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\,\mathrm{Cat}\,}}_n(q,t)$$\end{document} enumerates integer partitions contained in an n×n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n\times n$$\end{document} 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 Catn(q,t)=Catn(t,q)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\,\mathrm{Cat}\,}}_n(q,t)={{\,\mathrm{Cat}\,}}_n(t,q)$$\end{document} based on decomposing the set of all integer partitions into infinite chains. Each such global chain Cμ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {C}_{\mu }$$\end{document} has an opposite chain Cμ∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {C}_{\mu ^*}$$\end{document}; these combine to give a new small slice of Catn(q,t)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\,\mathrm{Cat}\,}}_n(q,t)$$\end{document} 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 Cμ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {C}_{\mu }$$\end{document} 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\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$11$$\end{document}. This proves that for all n, the terms in Catn(q,t)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\,\mathrm{Cat}\,}}_n(q,t)$$\end{document} of degree at least n2-11\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left( {\begin{array}{c}n\\ 2\end{array}}\right) -11$$\end{document} are symmetric in q and t.

Wang, George

doi: 10.1007/s00026-020-00513-4pmid: N/A

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.

Khosravi, Behnam; Khosravi, Behrooz; Khosravi, Bahman

doi: 10.1007/s00026-020-00514-3pmid: N/A

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.

Brenti, Francesco; Sentinelli, Paolo

doi: 10.1007/s00026-020-00515-2pmid: N/A

We define and study odd and even analogues of the major index statistics for the classical Weyl groups. More precisely, we show that the generating functions of these statistics, twisted by the one-dimensional characters of the corresponding groups, always factor in an explicit way. In particular, we obtain odd and even analogues of Carlitz’s identity, of the Gessel–Simion Theorem, and a parabolic extension, and refinement, of a result of Wachs.