Annals of Combinatorics最新文献

In 2009, Berkovich and Garvan introduced a new partition statistic called the GBG-rank modulo t which is a generalization of the well-known BG-rank. In this paper, we use the Littlewood decomposition of partitions to study partitions with bounded largest part and the fixed integral value of GBG-rank modulo primes. As a consequence, we obtain new elegant generating function formulas for unrestricted partitions, self-conjugate partitions, and partitions whose parts repeat a finite number of times.

A tangram is a word in which every letter occurs an even number of times. Such word can be cut into parts that can be arranged into two identical words. The minimum number of cuts needed is called the cut number of a tangram. For example, the word is a tangram with cut number one, while the word is a tangram with cut number two. Clearly, tangrams with cut number one coincide with the well-known family of words, known as squares, having the form UU for some nonempty word U. A word W avoids a word T if it is not possible to write (W=ATB), for any words A and B (possibly empty). The famous 1906 theorem of Thue asserts that there exist arbitrarily long words avoiding squares over an alphabet with just three letters. Given a fixed number (kgeqslant 1), how many letters are needed to avoid tangrams with the cut number at most k? Let t(k) denote the minimum size of an alphabet needed for that purpose. By Thue’s result we have (t(1)=3), which easily implies (t(2)=3). Curiously, these are currently the only known exact values of this function. In our main result we prove that (t(k)=Theta (log _2k)). The proof uses entropy compression argument and Zimin words. Using a different method we prove that (t(k)leqslant k+1) for all (kgeqslant 4), which gives more exact estimates for small values of k. The proof makes use of Dejean words and a curious property of Gauss words, which is perhaps of independent interest.

We revisit the twisted multiplicativity property of Voiculescu’s S-transform in the operator-valued setting, using a specific bijection between planar binary trees and noncrossing partitions.

A complete dessin is an orientable map with underlying graph being a complete bipartite graph, which is said to be regular if its group of color- and orientation-preserving automorphisms acts regularly on the edges. This paper presents a classification and enumeration of complete regular dessins that have exactly two faces, which is not only a generalization of uniface regular dessins (see Fan et al. in J Algebr Comb 49:125–134, 2019) and planar two-face maps (see Bousquet et al. in Discrete Math 222:1–25, 2000), but also a special case of the classification of complete regular dessins proposed by Jones.

Let (G otimes _f H) denote the Sierpiński product of graphs G and H with respect to the function f. The Sierpiński general position number (textrm{gp}{_{textrm{S}}}(G,H)) is introduced as the cardinality of a largest general position set in (G otimes _f H) over all possible functions f. Similarly, the lower Sierpiński general position number (underline{textrm{gp}}{_{textrm{S}}}(G,H)) is the corresponding smallest cardinality. The concept of vertex-colinear sets is introduced. Bounds for the general position number in terms of extremal vertex-colinear sets, and bounds for the (lower) Sierpiński general position number are proved. The extremal graphs are investigated. Formulas for the (lower) Sierpiński general position number of the Sierpiński products with (K_2) as the first factor are deduced. It is proved that if (m,nge 2), then (textrm{gp}{_{textrm{S}}}(K_m,K_n) = m(n-1)), and that if (nge 2,m-2), then (underline{textrm{gp}}{_{textrm{S}}}(K_m,K_n) = m(n-m+1)).

Plane partitions in the totally symmetric self-complementary symmetry class (TSSCPPs) are known to be equinumerous with (ntimes n) alternating sign matrices, but no explicit bijection is known. In this paper, we give a bijection from these plane partitions to ({0,1,-1})-matrices we call magog matrices, some of which are alternating sign matrices. We explore enumerative properties of these matrices related to natural statistics such as inversion number and number of negative ones. We then investigate the polytope defined as their convex hull. We show that all the magog matrices are extreme and give a partial inequality description. Finally, we define another TSSCPP polytope as the convex hull of TSSCPP boolean triangles and determine its dimension, inequalities, vertices, and facets.

We define a far-reaching generalization of Schnyder woods which encompasses many classical combinatorial structures on planar graphs. Schnyder woods are defined for planar triangulations as certain triples of spanning trees covering the triangulation and crossing each other in an orderly fashion. They are of theoretical and practical importance, as they are central to the proof that the order dimension of any planar graph is at most 3, and they are also underlying an elegant drawing algorithm. In this article, we extend the concept of Schnyder wood well beyond its original setting: for any integer (dge 3), we define a “grand-Schnyder” structure for (embedded) planar graphs which have faces of degree at most d and non-facial cycles of length at least d. We prove the existence of grand-Schnyder structures, provide a linear construction algorithm, describe 4 different incarnations (in terms of tuples of trees, corner labelings, weighted orientations, and marked orientations), and define a lattice for the set of grand-Schnyder structures of a given planar graph. We show that the grand-Schnyder framework unifies and extends several classical constructions: Schnyder woods and Schnyder decompositions, regular edge-labelings (a.k.a. transversal structures), and Felsner woods.

Estimating phylogenetic trees, which depict the relationships between different species, from aligned sequence data (such as DNA, RNA, or proteins) is one of the main aims of evolutionary biology. However, tree reconstruction criteria like maximum parsimony do not necessarily lead to unique trees and in some cases even fail to recognize the “correct” tree (i.e., the tree on which the data was generated). On the other hand, a recent study has shown that for an alignment containing precisely those binary characters (sites) which require up to two substitutions on a given tree, this tree will be the unique maximum parsimony tree. It is the aim of the present paper to generalize this recent result in the following sense: We show that for a tree T with n leaves, as long as (k<frac{n}{8}+frac{11}{9}-frac{1}{18}sqrt{9cdot left( frac{n}{4}right) ^2+16}) (or, equivalently, (n>9k-11+sqrt{9k^2-22k+17}), which in particular holds for all (nge 12k)), the maximum parsimony tree for the alignment containing all binary characters which require (up to or precisely) k substitutions on T will be unique in the NNI neighborhood of T and it will coincide with T, too. In other words, within the NNI neighborhood of T, T is the unique most parsimonious tree for the said alignment. This partially answers a recently published conjecture affirmatively. Additionally, we show that for (nge 8) and for k being in the order of (frac{n}{2}), there is always a pair of phylogenetic trees T and (T') which are NNI neighbors, but for which the alignment of characters requiring precisely k substitutions each on T in total requires fewer substitutions on (T').

Let A(p, n, k) be the number of p-tuples of commuting permutations of n elements whose permutation action results in exactly k orbits or connected components. We formulate the conjecture that, for every fixed p and n, the A(p, n, k) form a log-concave sequence with respect to k. For (p=1) this is a well-known property of unsigned Stirling numbers of the first kind. As the (p=2) case, our conjecture includes a previous one by Heim and Neuhauser, which strengthens a unimodality conjecture for the Nekrasov–Okounkov hook length polynomials. In this article, we prove the (p=infty ) case of our conjecture. We start from an expression for the A(p, n, k),  which follows from an identity by Bryan and Fulman, obtained in the their study of orbifold higher equivariant Euler characteristics. We then derive the (prightarrow infty ) asymptotics. The last step essentially amounts to the log-concavity in k of a generalized Turán number, namely, the maximum product of k positive integers whose sum is n.

It is known that the plane partition function of n denoted (textrm{PL}(n)) obeys Benford’s law in any integer base (bge 2). We give an upper bound for the smallest positive integer n such that (textrm{PL}(n)) starts with a prescribed string f of digits in base b.