Authors:Richard Ehrenborg; Alex Happ Abstract: We give a short proof that the f-vector of the descent polytope \({{\,\mathrm{DP}\,}}_{\mathbf {v}}\) is componentwise maximized when the word \(\mathbf {v}\) is alternating. Our proof uses an f-vector analog of the boustrophedon transform. PubDate: 2019-02-12 DOI: 10.1007/s00026-019-00422-1

Authors:Geoffrey Pearce; Cheryl E. Praeger Abstract: A graph is Cartesian decomposable if it is isomorphic to a Cartesian product of strictly smaller graphs, each of which has more than one vertex and admits no such decomposition. These smaller graphs are called the Cartesian-prime factors of the Cartesian decomposition, and were shown, by Sabidussi and Vizing independently, to be uniquely determined up to isomorphism. We characterise by their parameters those generalised Paley graphs which are Cartesian decomposable, and we prove that for such graphs, the Cartesian-prime factors are themselves smaller generalised Paley graphs. This generalises a result of Lim and the second author which deals with the case where all the Cartesian-prime factors are complete graphs. These results contribute to the determination, by parameters, of generalised Paley graphs with automorphism groups larger than the one-dimensional affine subgroups used to define them. They also shed light on the structure of primitive cyclotomic association schemes. PubDate: 2019-02-10 DOI: 10.1007/s00026-019-00423-0

Authors:Mingjia Yang Abstract: Enumeration problems related to words avoiding patterns as well as permutations that contain the pattern 123 exactly once have been studied in great detail. However, the problem of enumerating words that contain the pattern 123 exactly once is new and will be the focus of this paper. Previously, Zeilberger provided a shortened version of Burstein’s combinatorial proof of Noonan’s theorem which states that the number of permutations with exactly one 321 pattern is equal to \(\frac{3}{n} \left( {\begin{array}{c}2n\\ n+3\end{array}}\right) \) . Surprisingly, a similar method can be directly adapted to words. We are able to use this method to find a formula enumerating the words with exactly one 123 pattern. Further inspired by Shar and Zeilberger’s work on generating functions enumerating 123-avoiding words with r occurrences of each letter, we examine the algebraic equations for generating functions for words with r occurrences of each letter and with exactly one 123 pattern. PubDate: 2019-02-08 DOI: 10.1007/s00026-019-00416-z

Authors:Sami Mustapha Abstract: We prove that the sequence \(\left( e_n^{\Gamma }\right) _{n\in \mathbb {N}}\) of numbers of excursions in a three-quadrant cone corresponding to a non-singular step set \(\Gamma \subset \{0, \pm 1\}^2\) with an infinite group does not satisfy any nontrivial linear recurrence with polynomial coefficients. This gives a positive answer to a question asked by M. Bousquet-Mélou about D-finiteness of the trivariate generating function of the numbers of walks with given length and prescribed ending point. In the process, we determine the asymptotics of \(e_n^{\Gamma }\) , \(n\rightarrow \infty \) , for the 74 non-singular two-dimensional models, giving the first complete computation of excursions asymptotics in a non-convex cone. Moreover, using potential theoretic comparison arguments, we give the asymptotics of the number of walks avoiding the negative quadrant and of length n for all non-singular step sets having zero drift. PubDate: 2019-02-06 DOI: 10.1007/s00026-019-00413-2

Authors:Takuya Kusunoki; Satoshi Murai Abstract: A basic combinatorial invariant of a convex polytope P is its f-vector \(f(P)=(f_0,f_1,\dots ,f_{\dim P-1})\) , where \(f_i\) is the number of i-dimensional faces of P. Steinitz characterized all possible f-vectors of 3-polytopes and Grünbaum characterized the pairs given by the first two entries of the f-vectors of 4-polytopes. In this paper, we characterize the pairs given by the first two entries of the f-vectors of 5-polytopes. The same result was also proved by Pineda-Villavicencio, Ugon and Yost independently. PubDate: 2019-02-06 DOI: 10.1007/s00026-019-00417-y

Authors:Ezgi Kantarcı Oğuz Abstract: In 2015, Jing and Li defined type B quasisymmetric Schur functions and conjectured that these functions have a positive, integral and unitriangular expansion into peak functions. We prove this conjecture, and refine their combinatorial model to give explicit expansions in monomial, fundamental and peak bases. We also show that these functions are not quasisymmetric Schur, Young quasisymmetric Schur or dual immaculate positive, and do not have a positive multiplication rule. PubDate: 2019-02-04 DOI: 10.1007/s00026-019-00415-0

Authors:Oliver Roche-Newton; Igor E. Shparlinski; Arne Winterhof Abstract: Balog and Wooley have recently proved that any subset \({\mathcal {A}}\) of either real numbers or of a prime finite field can be decomposed into two parts \({\mathcal {U}}\) and \({\mathcal {V}}\) , one of small additive energy and the other of small multiplicative energy. In the case of arbitrary finite fields, we obtain an analogue that under some natural restrictions for a rational function f both the additive energies of \({\mathcal {U}}\) and \(f({\mathcal {V}})\) are small. Our method is based on bounds of character sums which leads to the restriction \(\# {\mathcal {A}}> q^{1/2}\) , where q is the field size. The bound is optimal, up to logarithmic factors, when \(\# {\mathcal {A}}\ge q^{9/13}\) . Using \(f(X)=X^{-1}\) we apply this result to estimate some triple additive and multiplicative character sums involving three sets with convolutions \(ab+ac+bc\) with variables a, b, c running through three arbitrary subsets of a finite field. PubDate: 2019-02-02 DOI: 10.1007/s00026-019-00420-3

Authors:Gene B. Kim Abstract: The distribution of descents in certain conjugacy classes of \(S_n\) has been previously studied, and it is shown that its moments have interesting properties. This paper provides a bijective proof of the symmetry of the descents and major indices of matchings (also known as fixed point free involutions) and uses a generating function approach to prove an asymptotic normality theorem for the number of descents in matchings. PubDate: 2019-02-01 DOI: 10.1007/s00026-019-00414-1

Authors:Benny Chor; Péter L. Erdős; Yonatan Komornik Abstract: Given two binary trees on N labeled leaves, the quartet distance between the trees is the number of disagreeing quartets. By permuting the leaves at random, the expected quartet distance between the two trees is \(\frac{2}{3}\left( {\begin{array}{c}N\\ 4\end{array}}\right) \) . However, no strongly explicit construction reaching this bound asymptotically was known. We consider complete, balanced binary trees on \(N=2^n\) leaves, labeled by n bits long sequences. Ordering the leaves in one tree by the prefix order, and in the other tree by the suffix order, we show that the resulting quartet distance is \(\left( \frac{2}{3} + o(1)\right) \left( {\begin{array}{c}N\\ 4\end{array}}\right) \) , and it always exceeds the \(\frac{2}{3}\left( {\begin{array}{c}N\\ 4\end{array}}\right) \) bound. PubDate: 2019-01-05 DOI: 10.1007/s00026-018-0411-3

Authors:Ross Atkins; Colin McDiarmid Abstract: Three standard subtree transfer operations for binary trees, used in particular for phylogenetic trees, are: tree bisection and reconnection (TBR), subtree prune and regraft (SPR), and rooted subtree prune and regraft (rSPR). We show that for a pair of leaf-labelled binary trees with n leaves, the maximum number of such moves required to transform one into the other is \(n-\Theta (\sqrt{n})\) , extending a result of Ding, Grünewald, and Humphries, and this holds also if one of the trees is fixed arbitrarily. If the pair is chosen uniformly at random, then the expected number of moves required is \(n-\Theta (n^{2/3})\) . These results may be phrased in terms of agreement forests: we also give extensions for more than two binary trees. PubDate: 2018-12-13 DOI: 10.1007/s00026-018-0410-4

Authors:Brian Cook; Ákos Magyar; Tatchai Titichetrakun Abstract: Let A be a subset of positive relative upper density of \(\mathbb {P}^d\) , the d-tuples of primes. We present an essentially self-contained, combinatorial argument to show that A contains infinitely many affine copies of any finite set \(F\subseteq \mathbb {Z}^d\) . This provides a natural multidimensional extension of the theorem of Green and Tao on the existence of long arithmetic progressions in the primes. PubDate: 2018-11-30 DOI: 10.1007/s00026-018-0402-4

Authors:Victor Reiner; Eric Sommers Abstract: Catalan numbers are known to count noncrossing set partitions, while Narayana and Kreweras numbers refine this count according to the number of blocks in the set partition, and by its collection of block sizes. Motivated by reflection group generalizations of Catalan numbers and their q-analogues, this paper concerns a definition of q-Kreweras numbers for finite Weyl groups W, refining the q-Catalan numbers for W, and arising from work of the second author. We give explicit formulas in all types for the q-Kreweras numbers. In the classical types A, B, C, we also record formulas for the q-Narayana numbers and in the process show that the formulas depend only on the Weyl group (that is, they coincide in types B and C). In addition, we verify that in the classical types A, B, C, D the q-Kreweras numbers obey the expected cyclic sieving phenomena when evaluated at appropriate roots of unity. PubDate: 2018-11-13 DOI: 10.1007/s00026-018-0408-y

Authors:Louis Gaudet; David Jensen; Dhruv Ranganathan; Nicholas Wawrykow; Theodore Weisman Abstract: We study which groups with pairing can occur as the Jacobian of a finite graph. We provide explicit constructions of graphs whose Jacobian realizes a large fraction of odd groups with a given pairing. Conditional on the generalized Riemann hypothesis, these constructions yield all groups with pairing of odd order, and unconditionally, they yield all groups with pairing whose prime factors are sufficiently large. For groups with pairing of even order, we provide a partial answer to this question, for a certain restricted class of pairings. Finally, we explore which finite abelian groups occur as the Jacobian of a simple graph. There exist infinite families of finite abelian groups that do not occur as the Jacobians of simple graphs. PubDate: 2018-11-02 DOI: 10.1007/s00026-018-0406-0

Authors:Mark Wildon Abstract: This paper proves a combinatorial rule expressing the product \(s_\tau (s_{\lambda /\mu } \circ p_r)\) of a Schur function and the plethysm of a skew Schur function with a power-sum symmetric function as an integral linear combination of Schur functions. This generalizes the SXP rule for the plethysm \(s_\lambda \circ p_r\) . Each step in the proof uses either an explicit bijection or a sign-reversing involution. The proof is inspired by an earlier proof of the SXP rule due to Remmel and Shimozono (Discrete Math. 193:257–266, 1998). The connections with two later combinatorial rules for special cases of this plethysm are discussed. Two open problems are raised. The paper is intended to be readable by non-experts. PubDate: 2018-11-01 DOI: 10.1007/s00026-018-0409-x

Authors:Byungchan Kim; Pee Choon Toh Abstract: We study a crank function M(m, n) for cubic partition pairs. We show that the function M(m, n) explains a cubic partition pair congruence and we also obtain various arithmetic properties regarding M(m, n). In particular, using the \(\Theta \) -operator, we confirm a conjecture on the sign pattern of c(n), the number of cubic partition pairs of n, weighted by the parity of the crank. PubDate: 2018-10-13 DOI: 10.1007/s00026-018-0407-z

Authors:Richard Ehrenborg Abstract: We study how the shelling components behave under the pyramid and prism operations. As a consequence we obtain a concise recursion for the cubical shelling contributions. PubDate: 2018-10-11 DOI: 10.1007/s00026-018-0412-2

Authors:Raymond Cheng; David M. Jackson; Geoff J. Stanley Abstract: Quasi-triangular Hopf algebras were introduced by Drinfel’d in his construction of solutions to the Yang–Baxter Equation. This algebra is built upon \(\mathscr {U}_h(\mathfrak {sl}_2)\) , the quantized universal enveloping algebra of the Lie algebra \(\mathfrak {sl}_2\) . In this paper, combinatorial structure in \(\mathscr {U}_h(\mathfrak {sl}_2)\) is elicited, and used to assist in highly intricate calculations in this algebra. To this end, a combinatorial methodology is formulated for straightening algebraic expressions to a canonical form in the case \(n=1\) . We apply this formalism to the quasi-triangular Hopf algebras and obtain a constructive account not only for the derivation of the Drinfel’d’s \(sR\) -matrix, but also for the arguably mysterious ribbon elements of \(\mathscr {U}_h(\mathfrak {sl}_2)\) . Finally, we extend these techniques to the higher-dimensional algebras \(\mathscr {U}_h(\mathfrak {sl}_{n+1})\) . While these explicit algebraic results are well known, our contribution is in our formalism and perspective: our emphasis is on the combinatorial structure of these algebras and how that structure may guide algebraic constructions. PubDate: 2018-10-10 DOI: 10.1007/s00026-018-0404-2

Authors:Noga Alon; Steve Butler; Ron Graham; Utkrisht C. Rajkumar Abstract: Let \(M = (s_1, s_2, \ldots , s_n) \) be a sequence of distinct symbols and \(\sigma \) a permutation of \(\{1,2, \ldots , n\}\) . Denote by \(\sigma (M)\) the permuted sequence \((s_{\sigma (1)}, s_{\sigma (2)}, \ldots , s_{\sigma (n)})\) . For a given positive integer d, we will say that \(\sigma \) is d-resilient if no matter how d entries of M are removed from M to form \(M'\) and d entries of \(\sigma (M)\) are removed from \(\sigma (M)\) to form \(\sigma (M)'\) (with no symbol being removed from both sequences), it is always possible to reconstruct the original sequence M from \(M'\) and \(\sigma (M)'\) . Necessary and sufficient conditions for a permutation to be d-resilient are established in terms of whether certain auxiliary graphs are acyclic. We show that for d-resilient permutations for [n] to exist, n must have size at least exponential in d, and we give an algorithm to construct such permutations in this case. We show that for each d and all sufficiently large n, the fraction of all permutations on n elements which are d-resilient is bounded away from 0. PubDate: 2018-10-05 DOI: 10.1007/s00026-018-0403-3

Authors:Akiyoshi Tsuchiya Abstract: Let \((P,\le _P)\) and \((Q,\le _Q)\) be finite partially ordered sets with \( P = Q =d\) , and \(\mathcal {C}(P) \subset \mathbb {R}^d\) and \(\mathcal {C}(Q) \subset \mathbb {R}^d\) their chain polytopes. The twinned chain polytope of P and Q is the lattice polytope \(\Gamma (\mathcal {C}(P),\mathcal {C}(Q)) \subset \mathbb {R}^d\) which is the convex hull of \(\mathcal {C}(P) \cup (-\mathcal {C}(Q))\) . It is known that twinned chain polytopes are Gorenstein Fano polytopes with the integer decomposition property. In the present paper, we study combinatorial properties of twinned chain polytopes. First, we will give the formula of the volume of twinned chain polytopes in terms of the underlying partially ordered sets. Second, we will identify the facet-supporting hyperplanes of twinned chain polytopes in terms of the underlying partially ordered sets. Finally, we will provide the vertex representations of the dual polytopes of twinned chain polytopes. PubDate: 2018-10-05 DOI: 10.1007/s00026-018-0405-1

Abstract: In this paper, we provide combinatorial proofs for certain partition identities which arise naturally in the context of Langlands’ beyond endoscopy proposal. These partition identities motivate an explicit plethysm expansion of \({{\rm Sym}^j}\) \({{{\rm Sym}^{k}}V}\) for \({{\rm GL}_2}\) in the case k = 3. We compute the plethysm explicitly for the cases k = 3, 4. Moreover, we use these expansions to explicitly compute the basic function attached to the symmetric power L-function of \({{\rm GL}_2}\) for these two cases. PubDate: 2018-06-06 DOI: 10.1007/s00026-018-0391-3