We show that powersets over structures with a bounded counting sequence can be sampled efficiently without evaluating the generating function. An algorithm is provided, implemented, and tested. Runtimes are comparable to existing Boltzmann samplers reported in the literature.
We characterize the orderings of pairs of sets induced by several distances: Hamming, Jaccard, Sørensen-Dice and Overlap. We also characterize these distances.
In this work we establish local limit theorems for q-multinomial and multiple Heine distributions. Specifically, the pointwise convergence of the q-multinomial distribution of the first kind, as well as for its discrete limit, the multiple Heine distribution, to a multivariate Stieltjes-Wigert type distribution, are provided.
We consider sorting procedures for permutations making use of pop stacks with a bypass operation, and explore the combinatorial properties of the associated algorithms.
We present three simple algorithms to uniformly generate `Fibonacci words' (i.e., some words that are enumerated by Fibonacci numbers), Schr{ö}der trees of size $n$ and Motzkin left factors of size $n$ and final height $h$. These algorithms have an average complexity of $O(n)$ in the unit-cost RAM model.
We show a slightly simpler proof the following theorem by I. Dinur, O. Regev, and C. Smyth: for all $c \geq 2$, it is NP-hard to find a $c$-colouring of a 2-coloruable 3-uniform hypergraph. We recast this result in the algebraic framework for Promise CSPs, using only a weaker version of the PCP theorem.
The original knapsack problem is well known to be NP-complete. In a multidimensional version one have to decide whether a $p\in \N^k$ is in a sumset-sum of a set $X \subseteq \N^k$ or not. In this paper we are going to investigate a communication complexity problem related to this.
We generalize Axel Thue's familiar definition of overlaps in words, and show that there are no infinite words containing split occurrences of these generalized overlaps. Along the way we prove a useful theorem about repeated disjoint occurrences in words -- an interesting natural variation on the classical de Bruijn sequences.
We show that the problem of deciding whether a given Euclidean lattice L has an orthonormal basis is in NP and co-NP. Since this is equivalent to saying that L is isomorphic to the standard integer lattice, this problem is a special form of the Lattice Isomorphism Problem, which is known to be in the complexity class SZK.
This paper proposes an algebraic view of trees which opens the doors to an alternative computational scheme with respect to classic algorithms. In particular, it is shown that this view is very well-suited for machine learning and computational linguistics.
El artículo presenta la llegada del nuevo milenio, un número cada vez mayor de empresarios se unieron a la aplicación del diseño sostenible que comenzó a replantearse en las empresas y el rol que juegan con el desarrollo del medio ambiente, el planeta y en la sociedad. Podemos decir que el diseño sostenible busca generar soluciones a través de servicios y estilos de vida, pero no exclusivamente a través de objetos. Con el fin de introducir una definición elaborada de diseño sostenible es necesario mencionar los sistemas sostenibles, que básicamente, se refieren a cualquier tipo de red o servicio social que puede existir y replicarse. Además de sistemas sostenibles hay otros principios dentro del diseño sostenible. Por último, cualquier tipo de resultado obtenido para satisfacer la necesidad debe ser sostenible a largo plazo entendiéndose como un proceso que permita una comunidad lograr un resultado a través de estrategias de diseño.
There is a sizable literature on investigating the minimum and maximum numbers of cycles in a class of graphs. However, the answer is known only for special classes. This paper presents a result on the smallest number of cycles in hamiltonian 3-connected cubic graphs. Further, it describes a proof technique that could improve an upper bound of the largest number of cycles in a hamiltonian graph.
We derive combinatorial identities for variables satisfying specific sets of commutation relations. The identities thus obtained extend corresponding ones for $q$-commuting variables $x$ and $y$ satisfying $yx=qxy$. In particular, we obtain weight-dependent binomial theorems, functional equations for generalized exponential functions, we propose a derivative of noncommuting variables, and finally utilize one of the considered weight functions to extend rook theory. This leads us to an extension of the $q$-Stirling numbers of the second kind, and of the $q$-Lah numbers.
A permutation $σ$ contains the permutation $τ$ if there is a subsequence of $σ$ order isomorphic to $τ$. A permutation $σ$ is $τ$-<i>avoiding</i> if it does not contain the permutation $τ$. For any $n$, the <i>popularity</i> of a permutation $τ$, denoted $A$<sub>$n$</sub>($τ$), is the number of copies of $τ$ contained in the set of all 132-avoiding permutations of length $n$. Rudolph conjectures that for permutations $τ$ and $μ$ of the same length, $A$<sub>$n$</sub>($τ$) ≤ $A$<sub>$n$</sub>($μ$) for all $n$ if and only if the spine structure of $τ$ is less than or equal to the spine structure of $μ$ in refinement order. We prove one direction of this conjecture, by showing that if the spine structure of $τ$ is less than or equal to the spine structure of $μ$, then $A$<sub>$n$</sub>($τ$) ≤ $A$<sub>$n$</sub>($μ$) for all $n$. We disprove the opposite direction by giving a counterexample, and hence disprove the conjecture.
A spanning subgraph $F$ of a graph $G$ is called perfect if $F$ is a forest, the degree $d_F(x)$ of each vertex $x$ in $F$ is odd, and each tree of $F$ is an induced subgraph of $G$. We provide a short proof of the following theorem of A.D. Scott (Graphs & Combin., 2001): a connected graph $G$ contains a perfect forest if and only if $G$ has an even number of vertices.
We define an action of the $0$-Hecke algebra of type A on the Stanley-Reisner ring of the Boolean algebra. By studying this action we obtain a family of multivariate noncommutative symmetric functions, which specialize to the noncommutative Hall-Littlewood symmetric functions and their $(q,t)$-analogues introduced by Bergeron and Zabrocki. We also obtain multivariate quasisymmetric function identities, which specialize to a result of Garsia and Gessel on the generating function of the joint distribution of five permutation statistics.
O'Donnell, Wright, Wu and Zhou [SODA 2014] introduced the notion of robustly asymmetric graphs. Roughly speaking, these are graphs in which for every $0 \le ρ\le 1$, every permutation that permutes a $ρ$ fraction of the vertices maps a $Θ(ρ)$ fraction of the edges to non-edges. We show that there are graphs for which the constant hidden in the $Θ$ notation is roughly~1.
We develop a new method for studying the asymptotics of symmetric polynomials of representation–theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems concerning characters of infinite–dimensional unitary group and their $q$–deformations. We study the behavior of uniformly random lozenge tilings of large polygonal domains and find the GUE–eigenvalues distribution in the limit. We also investigate similar behavior for Alternating Sign Matrices (equivalently, six–vertex model with domain wall boundary conditions). Finally, we compute the asymptotic expansion of certain observables in the $O(n=1)$ dense loop model.
We show the $\mathrm{cd}$-index exists for Whitney stratified manifolds by extending the notion of a graded poset to that of a quasi-graded poset. This is a poset endowed with an order-preserving rank function and a weighted zeta function. This allows us to generalize the classical notion of Eulerianness, and obtain a $\mathrm{cd}$-index in the quasi-graded poset arena. We also extend the semi-suspension operation to that of embedding a complex in the boundary of a higher dimensional ball and study the shelling components of the simplex.