Annals of Combinatorics最新文献

We generalize valuations on polyhedral cones to valuations on (plane) fans. For fans induced by hyperplane arrangements, we show a correspondence between rotation-invariant valuations and deletion– restriction invariants. In particular, we define a characteristic polynomial for fans in terms of spherical intrinsic volumes and show that it coincides with the usual characteristic polynomial in the case of hyperplane arrangements. This gives a simple deletion–restriction proof of a result of Klivans–Swartz. The metric projection of a cone is a piecewise-linear map, whose underlying fan prompts a generalization of spherical intrinsic volumes to indicator functions. We show that these intrinsic indicators yield valuations that separate polyhedral cones. Applied to hyperplane arrangements, this generalizes a result of Kabluchko on projection volumes.

Given a graph X with a Hamilton cycle C, the compression factor (kappa (X,C)) of C is the order of the largest cyclic subgroup of ({textrm{Aut}},(C)cap {textrm{Aut}},(X)), and the Hamilton compression (kappa (X)) of X is the maximum of (kappa (X,C)) where C runs over all Hamilton cycles in X. Generalizing the well-known open problem regarding the existence of vertex-transitive graphs without Hamilton paths/cycles, it was asked by Gregor et al. (Ann Comb, arXiv:2205.08126v1, https://doi.org/10.1007/s00026-023-00674-y, 2023) whether for every positive integer k, there exists infinitely many vertex-transitive graphs (Cayley graphs) with Hamilton compression equal to k. Since an infinite family of Cayley graphs with Hamilton compression equal to 1 was given there, the question is completely resolved in this paper in the case of Cayley graphs with a construction of Cayley graphs of semidirect products (mathbb {Z}_prtimes mathbb {Z}_k) where p is a prime and (k ge 2) a divisor of (p-1). Further, infinite families of non-Cayley vertex-transitive graphs with Hamilton compression equal to 1 are given. All of these graphs being metacirculants, some additional results on Hamilton compression of metacirculants of specific orders are also given.

The B-polynomial and quasisymmetric B-function, introduced by Awan and Bernardi, extends the widely studied Tutte polynomial and Tutte symmetric function to digraphs. In this article, we address one of the fundamental questions concerning these digraph invariants, which is, the determination of the classes of digraphs uniquely characterized by them. We solve an open question originally posed by Awan and Bernardi, regarding the identification of digraphs that result from replacing every edge of a graph with a pair of opposite arcs. Further, we address the more challenging problem of reconstructing digraphs using their quasisymmetric functions. In particular, we show that the quasisymmetric B-function reconstructs partially symmetric orientations of proper caterpillars. As a consequence, we establish that all orientations of paths and asymmetric proper caterpillars can be reconstructed from their quasisymmetric B-functions. These results enhance the pool of oriented trees distinguishable through quasisymmetric functions.

In recent literature concerning integer partitions one can find many results related to both the Bessenrodt–Ono type inequalities and log-concavity properties. In this note, we offer some general approach to this type of problems. More precisely, we prove that under some mild conditions on an increasing function F of at most exponential growth satisfying the condition (F(mathbb {N})subset mathbb {R}_{+}), we have (F(a)F(b)>F(a+b)) for sufficiently large positive integers a, b. Moreover, we show that if the sequence ((F(n))_{nge n_{0}}) is log-concave and (limsup _{nrightarrow +infty }F(n+n_{0})/F(n)<F(n_{0})), then F satisfies the Bessenrodt–Ono type inequality.

Multidimensional permutations, or d-permutations, are represented by their diagrams on ([n]^d) such that there exists exactly one point per hyperplane (x_i) that satisfies (x_i= j) for (i in [d]) and (j in [n]). Bonichon and Morel previously enumerated 3-permutations avoiding small patterns, and we extend their results by first proving four conjectures, which exhaustively enumerate 3-permutations avoiding any two fixed patterns of size 3. We further provide a enumerative result relating 3-permutation avoidance classes with their respective recurrence relations. In particular, we show a recurrence relation for 3-permutations avoiding the patterns 132 and 213, which contributes a new sequence to the OEIS database. We then extend our results to completely enumerate 3-permutations avoiding three patterns of size 3.

The plethystic Murnaghan–Nakayama rule describes how to decompose the product of a Schur function and a plethysm of the form (p_rcirc h_m) as a sum of Schur functions. We provide a short, entirely combinatorial proof of this rule using the labelled abaci introduced in Loehr (SIAM J Discrete Math 24(4):1356–1370, 2010).

Two d-dimensional simplices in (mathbb {R}^d) are neighborly if its intersection is a ((d-1))-dimensional set. A family of d-dimensional simplices in (mathbb {R}^d) is called neighborly if every two simplices of the family are neighborly. Let (S_d) be the maximal cardinality of a neighborly family of d-dimensional simplices in (mathbb {R}^d). Based on the structure of some codes (Vsubset {0,1,*}^n) it is shown that (lim _{drightarrow infty }(2^{d+1}-S_d)=infty ). Moreover, a result on the structure of codes (Vsubset {0,1,*}^n) is given.

Let s be West’s stack-sorting map, and let (s_{T}) be the generalized stack-sorting map, where instead of being required to increase, the stack avoids subpermutations that are order-isomorphic to any permutation in the set T. In 2020, Cerbai, Claesson, and Ferrari introduced the (sigma )-machine (s circ s_{sigma }) as a generalization of West’s 2-stack-sorting-map (s circ s). As a further generalization, in 2021, Baril, Cerbai, Khalil, and Vajnovski introduced the ((sigma , tau ))-machine (s circ s_{sigma , tau }) and enumerated (textrm{Sort}_{n}(sigma ,tau ))—the number of permutations in (S_n) that are mapped to the identity by the ((sigma , tau ))-machine—for six pairs of length 3 permutations ((sigma , tau )). In this work, we settle a conjecture by Baril, Cerbai, Khalil, and Vajnovski on the only remaining pair of length 3 patterns ((sigma , tau ) = (132, 321)) for which (|textrm{Sort}_{n}(sigma , tau )|) appears in the OEIS. In addition, we enumerate (textrm{Sort}_n(123, 321)), which does not appear in the OEIS, but has a simple closed form.

We introduce a symmetry class for higher dimensional partitions—fully complementary higher dimensional partitions (FCPs)—and prove a formula for their generating function. By studying symmetry classes of FCPs in dimension 2, we define variations of the classical symmetry classes for plane partitions. As a by-product, we obtain conjectures for three new symmetry classes of plane partitions and prove that another new symmetry class, namely quasi-transpose-complementary plane partitions, are equinumerous to symmetric plane partitions.

We define and construct the “canonical reduced word” of a boolean permutation, and show that the RSK tableaux for that permutation can be read off directly from this reduced word. We also describe those tableaux that can correspond to boolean permutations, and enumerate them. In addition, we generalize a result of Mazorchuk and Tenner, showing that the “run” statistic influences the shape of the RSK tableau of arbitrary permutations, not just of those that are boolean.