Annals of Combinatorics

Berndt, Bruce; Zhou, Roberta

doi: 10.1007/s00026-015-0273-xpmid: N/A

We establish several identities for partitions with distinct colors that arise from Ramanujan’s formulas for multipliers in the theory of modular equations. In the course of our investigations, we prove three conjectures made by C. Sandon and F. Zanello. However, combinatorial proofs of these three conjectures and most of the theorems in this paper remain to be given.

Constantine, Gregory

doi: 10.1007/s00026-015-0274-9pmid: N/A

For any prime power q we construct a nonlinear code consisting of 8q codewords, each codeword having length 4q, and with minimum distance 2(q − 1). When compared to a Hadamard code of the same size it is shown that the new code corrects at most one error less, irrespective of q. Hadamard codes, however, are not known to always exist, whereas the new codes exist always.

Jing, Naihuan; Nie, Benzhi

doi: 10.1007/s00026-015-0271-zpmid: N/A

Vertex operator realizations of symplectic and orthogonal Schur functions are studied and expanded. New proofs of determinant identities of irreducible characters for the symplectic and orthogonal groups are given. We also give a new proof of the duality between the universal orthogonal and symplectic Schur functions using vertex operators.

Josuat-Vergès, Matthieu

doi: 10.1007/s00026-015-0278-5pmid: N/A

In the combinatorics of finite Coxeter groups, there is a simple formula giving the number of maximal chains of noncrossing partitions. It is a reinterpretation of a result by Deligne which is due to Chapoton, and the goal of this article is to refine the formula. First, we prove a one-parameter generalization, by considering the enumeration of noncrossing chains where we put a weight on some relations. Second, we consider an equivalence relation on noncrossing chains coming from the natural action of the group on set partitions, and we show that each equivalence class has a simple generating function. Using this we recover Postnikov’s hook length formula in type A and obtain a variant in type B.

Kurkova, I.; Raschel, K.

doi: 10.1007/s00026-015-0279-4pmid: N/A

In this article we obtain new expressions for the generating functions counting (nonsingular) walks with small steps in the quarter plane. Those are given in terms of infinite series, while in the literature, the standard expressions use solutions to boundary value problems. We illustrate our results with three examples (an algebraic case, a transcendental D-finite case, and an infinite group model).

Louis, Justine

doi: 10.1007/s00026-015-0272-ypmid: N/A

In this paper we obtain precise asymptotics for certain families of graphs, namely circulant graphs and degenerating discrete tori. The asymptotics contain interesting constants from number theory among which some can be interpreted as corresponding values for continuous limiting objects. We answer one question formulated in a paper from Atajan, Yong, and Inaba in [1] and formulate a conjecture in relation to the paper from Zhang, Yong, and Golin [23]. A crucial ingredient in the proof is to use the matrix tree theorem and express the combinatorial Laplacian determinant in terms of Bessel functions. A non-standard Poisson summation formula and limiting properties of theta functions are then used to evaluate the asymptotics.

Manstavičius, Eugenijus

doi: 10.1007/s00026-015-0275-8pmid: N/A

Asymptotic probabilities of decomposable combinatorial structures with a fortiori prescribed properties are studied as the sizes unboundedly increase. A comparative analysis of the power series coefficients is applied.

Munagi, Augustine

doi: 10.1007/s00026-015-0276-7pmid: N/A

We present new proofs of certain partition identities which appeared in three published articles of Ashok K. Agarwal. The main tool is obtained by combining zig-zag graphs with their geometric connection to partition Ferrers graphs as hinted by MacMahon in his classic text Combinatory Analysis. We give combinatorial proofs of the identities and provide a generalization or refinement of each result.

Parkinson, James; Temmermans, Beukje; Van Maldeghem, Hendrik

doi: 10.1007/s00026-015-0277-6pmid: N/A

We investigate the combinatorial interplay between automorphisms and opposition in (primarily finite) generalised polygons. We provide restrictions on the fixed element structures of automorphisms of a generalised polygon mapping no chamber to an opposite chamber. Furthermore, we give a complete classification of automorphisms of finite generalised polygons which map at least one point and at least one line to an opposite, but map no chamber to an opposite chamber. Finally, we show that no automorphism of a finite thick generalised polygon maps all chambers to opposite chambers, except possibly in the case of generalised quadrangles with coprime parameters.