We enumerate injectively k-colored rooted forests with a given number of vertices of each color and a given sequence of root colors. We obtain from this result some new multi-parameter distributions of Fuss-Catalan nu...
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We enumerate injectively k-colored rooted forests with a given number of vertices of each color and a given sequence of root colors. We obtain from this result some new multi-parameter distributions of Fuss-Catalan numbers. As an additional application we enumerate triangulations of regular convex polygons according to their proper 3-coloring type.
We have studied self-avoiding walks contained within an L x L square whose end-points can lie anywhere within, or on, the boundaries of the square. We prove that such walks behave, asymptotically, as walks crossing a ...
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We have studied self-avoiding walks contained within an L x L square whose end-points can lie anywhere within, or on, the boundaries of the square. We prove that such walks behave, asymptotically, as walks crossing a square (WCAS), being those walks whose end-points lie at the south-east and northwest corners of the square. We provide numerical data, enumerating all such walks, and analyse the sequence of coefficients in order to estimate the asymptotic behaviour. We also studied a subset of these walks, those that must contain at least one edge on all four boundaries of the square. We provide compelling evidence that these two classes of walks grow identically. From our analysis we conjecture that the number of such walks C-L, for both problems, behaves as C-L similar to lambda(L2+bL+c).L-g, where (Guttmann and Jensen 2022 J. Phys. A: Math. Theor.) lambda = 1.744 5498 +/- 0.000 0012, b = -0.043 54 +/- 0.0005, c = -1.35 +/- 0.45, and g = 3.9 +/- 0.1. Finally. we also studied the equivalent problem for self-avoiding polygons, also known as cycles in a square grid. The asymptotic behaviour of cycles has the same form as walks, but with different values of the parameters c, and g. Our numerical analysis shows that lambda and b have the same values as for WCAS and that c 1.776 +/- 0.002 while g = -0.500 +/- 0.005 and hence probably equals - 1/2.
In this work, we generalize a recursive enumerative formula for connected Feynman diagrams with two external legs. The Feynman diagrams are defined from a fermionic gas with a two-body interaction. The generalized rec...
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In this work, we generalize a recursive enumerative formula for connected Feynman diagrams with two external legs. The Feynman diagrams are defined from a fermionic gas with a two-body interaction. The generalized recurrence is valid for connected Feynman diagrams with an arbitrary number of external legs and an arbitrary order. The recurrence formula terms are expressed in function of weak compositions of non-negative integers and partitions of positive integers in such a way that to each term of the recurrence correspond a partition and a weak composition. The foundation of this enumeration is the Wick theorem, permitting an easy generalization to any quantum field theory. The iterative enumeration is constructive and enables a fast computation of the number of connected Feynman diagrams for a large amount of cases. In particular, the recurrence is solved exactly for two and four external legs, leading to the asymptotic expansion of the number of different connected Feynman diagrams.
We present some results on the proportion of permutations of length n containing certain mesh patterns as n grows large, and give exact enumeration results in some cases. In particular, we focus on mesh patterns where...
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We present some results on the proportion of permutations of length n containing certain mesh patterns as n grows large, and give exact enumeration results in some cases. In particular, we focus on mesh patterns where entire rows and columns are shaded. We prove some general results which apply to mesh patterns of any length, and then consider mesh patterns of length four. An important consequence of these results is to show that the proportion of permutations containing a mesh pattern can take a wide range of values between 0 and 1. (c) 2022 The Author(s). Published by Elsevier *** is an open access article under the CC BY license (http://***/licenses/by/4.0/).
We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for undirected graphs. We prove that computing undirected determinants as well as permanents for planar grap...
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We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for undirected graphs. We prove that computing undirected determinants as well as permanents for planar graphs whose vertices have degree at most 4 is #P-complete. In the case of planar graphs whose vertices have degree at most 3, the computation of the undirected determinant remains #P-complete while computing the permanent can be reduced to the FKT algorithm, and therefore can be done in polynomial time. Computing the undirected permanent is a Holant problem and its complexity can be deduced from the existing literature. It is mentioned in the paper as a natural context but no new results in this direction are obtained. The concept of undirected determinant is new. Its introduction is motivated by the formal resemblance to the directed determinant, a property that may inspire generalizations of some of the many algorithms which compute the latter. For a sizable class of planar 3-regular graphs, we are able to compute the undirected determinant in polynomial time.
作者：
Zakeri, SaeedCUNY Queens Coll
Dept Math 65-30 Kissena Blvd Queens NY 11367 USA CUNY
Grad Ctr 365 Fifth Ave New York NY 10016 USA
This note will give an enumeration of n-cyclesin the symmetric group S-n by their degree (also known as their cyclic descent number) and studies similar counting problems for the conjugacy classes of n-cycles under th...
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This note will give an enumeration of n-cyclesin the symmetric group S-n by their degree (also known as their cyclic descent number) and studies similar counting problems for the conjugacy classes of n-cycles under the action of the rotation subgroup of S-n. This is achieved by relating such cycles to periodic orbits of an associated dynamical system acting on the circle. We also compute the mean and variance of the degree of a random n-cycle and show that its distribution is asymptotically normal as n -> 8. (C) 2021 Elsevier Inc. All rights reserved.
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