A versatile combinatorial approach of studying products of long cycles in symmetric groups

In symmetric groups, studies of permutation factorizations or triples of permutations satisfying certain conditions have a long history. One particular interesting case is when two of the involved permutations are long cycles, for which many surprisingly simple formulas have been obtained. Here we combinatorially enumerate the pairs of long cycles whose product has a given cycle-type and separates certain elements, extending several lines of studies, and we obtain general quantitative relations. As consequences, in a unified way, we recover a number of results expecting simple combinatorial proofs, including results of Boccara (1980), Zagier (1995), Stanley (2011), F\'{e}ray and Vassilieva (2012), as well as Hultman (2014). We obtain a number of new results as well. In particular, for the first time, given a partition of a set, we obtain an explicit formula for the number of pairs of long cycles on the set such that the product of the long cycles does not mix the elements from distinct blocks of the partition and has an independently prescribed number of cycles for each block of elements. As applications, we obtain new explicit formulas concerning factorizations of any even permutation into long cycles and the first nontrivial explicit formula for computing strong separation probabilities solving an open problem of Stanley (2010).Comment: 12 pages, a draft extended abstract, comments are welcome. arXiv admin note: substantial text overlap with arXiv:1909.13388; text overlap with arXiv:1910.0102

Efficient generation of random derangements with the expected distribution of cycle lengths

We show how to generate random derangements efficiently by two different techniques: random restricted transpositions and sequential importance sampling. The algorithm employing restricted transpositions can also be used to generate random fixed-point-free involutions only, a.k.a. random perfect matchings on the complete graph. Our data indicate that the algorithms generate random samples with the expected distribution of cycle lengths, which we derive, and for relatively small samples, which can actually be very large in absolute numbers, we argue that they generate samples indistinguishable from the uniform distribution. Both algorithms are simple to understand and implement and possess a performance comparable to or better than those of currently known methods. Simulations suggest that the mixing time of the algorithm based on random restricted transpositions (in the total variance distance with respect to the distribution of cycle lengths) is $O(n^{a}\log{n}^{2})$ with $a \simeq \frac{1}{2}$ and $n$ the length of the derangement. We prove that the sequential importance sampling algorithm generates random derangements in $O(n)$ time with probability $O(1/n)$ of failing.Comment: This version corrected and updated; 14 pages, 2 algorithms, 2 tables, 4 figure

Constructing Fully Complete Models of Multiplicative Linear Logic

The multiplicative fragment of Linear Logic is the formal system in this family with the best understood proof theory, and the categorical models which best capture this theory are the fully complete ones. We demonstrate how the Hyland-Tan double glueing construction produces such categories, either with or without units, when applied to any of a large family of degenerate models. This process explains as special cases a number of such models from the literature. In order to achieve this result, we develop a tensor calculus for compact closed categories with finite biproducts. We show how the combinatorial properties required for a fully complete model are obtained by this glueing construction adding to the structure already available from the original category.Comment: 72 pages. An extended abstract of this work appeared in the proceedings of LICS 201

The String Theory Approach to Generalized 2D Yang-Mills Theory

We calculate the partition function of the $SU(N)$ ( and $U(N)$) generalized $YM_2$ theory defined on an arbitrary Riemann surface. The result which is expressed as a sum over irreducible representations generalizes the Rusakov formula for ordinary YM_2 theory. A diagrammatic expansion of the formula enables us to derive a Gross-Taylor like stringy description of the model. A sum of 2D string maps is shown to reproduce the gauge theory results. Maps with branch points of degree higher than one, as well as ``microscopic surfaces'' play an important role in the sum. We discuss the underlying string theory.Comment: TAUP-2182-94, 53 pages of LaTeX and 5 uuencoded eps figure

Cutting and Shuffling a Line Segment: Mixing by Interval Exchange Transformations

We present a computational study of finite-time mixing of a line segment by cutting and shuffling. A family of one-dimensional interval exchange transformations is constructed as a model system in which to study these types of mixing processes. Illustrative examples of the mixing behaviors, including pathological cases that violate the assumptions of the known governing theorems and lead to poor mixing, are shown. Since the mathematical theory applies as the number of iterations of the map goes to infinity, we introduce practical measures of mixing (the percent unmixed and the number of intermaterial interfaces) that can be computed over given (finite) numbers of iterations. We find that good mixing can be achieved after a finite number of iterations of a one-dimensional cutting and shuffling map, even though such a map cannot be considered chaotic in the usual sense and/or it may not fulfill the conditions of the ergodic theorems for interval exchange transformations. Specifically, good shuffling can occur with only six or seven intervals of roughly the same length, as long as the rearrangement order is an irreducible permutation. This study has implications for a number of mixing processes in which discontinuities arise either by construction or due to the underlying physics.Comment: 21 pages, 10 figures, ws-ijbc class; accepted for publication in International Journal of Bifurcation and Chao

Tools for Quantum Algorithms

We present efficient implementations of a number of operations for quantum computers. These include controlled phase adjustments of the amplitudes in a superposition, permutations, approximations of transformations and generalizations of the phase adjustments to block matrix transformations. These operations generalize those used in proposed quantum search algorithms.Comment: LATEX, 15 pages, Minor changes: one author's e-mail and one reference numbe

The Gibbs Paradox Revisited

The Gibbs paradox has frequently been interpreted as a sign that particles of the same kind are fundamentally indistinguishable; and that quantum mechanics, with its identical fermions and bosons, is indispensable for making sense of this. In this article we shall argue, on the contrary, that analysis of the paradox supports the idea that classical particles are always distinguishable. Perhaps surprisingly, this analysis extends to quantum mechanics: even according to quantum mechanics there can be distinguishable particles of the same kind. Our most important general conclusion will accordingly be that the universally accepted notion that quantum particles of the same kind are necessarily indistinguishable rests on a confusion about how particles are represented in quantum theory.Comment: to appear in Proceedings of "The Philosophy of Science in a European Perspective 2009