Counting Subwords in a Partition of a Set
TL;DR: This paper finds simple explicit formulas for the total number of occurrences of the patterns in question within all the partitions of $[n]$ containing exactly $k$ blocks, providing both algebraic and combinatorial proofs.
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Abstract: A partition $\pi$ of the set $[n]=\{1,\ldots,n\}$ is a collection $\{B_1,\ldots ,B_k\}$ of nonempty disjoint subsets of $[n]$ (called blocks ) whose union equals $[n]$ In this paper, we find explicit formulas for the generating functions for the number of partitions of $[n]$ containing exactly $k$ blocks where $k$ is fixed according to the number of occurrences of a subword pattern $\tau$ for several classes of patterns, including all words of length 3 In addition, we find simple explicit formulas for the total number of occurrences of the patterns in question within all the partitions of $[n]$ containing $k$ blocks, providing both algebraic and combinatorial proofs
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Universal geometric cluster algebras
TL;DR: In this paper, a generalization of geometric cluster algebra over polyhedral geometry is proposed, where the universal object is defined as a universal geometric coefficient. And the universal geometric coefficients are constructed in finite and affine types.
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Universal geometric cluster algebras
TL;DR: In this article, for each exchange matrix B, a category of geometric cluster algebra over B and coefficient specializations between the cluster algebras are considered, and the universal geometric coefficients are constructed in rank 2 and in finite type and in affine type.
Counting subwords in flattened partitions of sets
TL;DR: In all cases explicit formulas and/or generating functions for the number of set partitions of size n which avoid a single subword pattern of length three are determined, and it is shown that theNumber of occurrences of the pattern asymptotically follows a normal distribution.
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Counting corners in compositions and set partitions presented as bargraphs
TL;DR: In this article, the authors consider statistics on compositions of a fixed number and set partitions of fixed size represented geometrically as bargraphs, and derive a joint distribution for corners of type uhi for all i less than some fixed number.
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Set partitions with circular successions
TL;DR: Methods include both elementary combinatorial reasoning and the application of ordinary and exponential power series generating functions for the enumeration of partitions of a finite set according to the number of consecutive elements inside a block under the assumption that the elements are arranged around a circle.
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References
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Enumerative Combinatorics
R P Stanley
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Abstract: Richard Stanley's two-volume basic introduction to enumerative combinatorics has become the standard guide to the topic for students and experts alike. This thoroughly revised second edition of volume two covers the composition of generating functions, in particular the exponential formula and the Lagrange inversion formula, labelled and unlabelled trees, algebraic, D-finite, and noncommutative generating functions, and symmetric functions. The chapter on symmetric functions provides the only available treatment of this subject suitable for an introductory graduate course and focusing on combinatorics, especially the Robinson–Schensted–Knuth algorithm. An appendix by Sergey Fomin covers some deeper aspects of symmetric functions, including jeu de taquin and the Littlewood–Richardson rule. The exercises in the book play a vital role in developing the material, and this second edition features over 400 exercises, including 159 new exercises on symmetric functions, all with solutions or references to solutions.
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Enumerative Combinatorics
Charalambos A. Charalambides
- 01 Jan 2002
TL;DR: This review of 3 Enumerative Combinatorics, by Charalambos A.good, does not support this; the label ‘Example’ is given in a rather small font followed by a ‘PROOF,’ and the body of an example is nonitalic, utterly unlike other statements accompanied by demonstrations.
3.1K
Hopf Algebras, Renormalization and Noncommutative Geometry
Alain Connes,Dirk Kreimer +1 more
TL;DR: In this paper, the relation between the Hopf algebra associated to the renormalization of QFT and Hopf algebras associated to NCG computations of tranverse index theory for foliations is explored.
1K
On the Hopf algebra structure of perturbative quantum field theories
TL;DR: In this article, it was shown that the process of renormalization encapsulates a Hopf algebra structure in a natural manner and sheds light on the recently proposed connection between knots and renormalisation theory.
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