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Recurrences for 2-colored and 3-colored F-partitions

James A. Sellers
1996 Discrete Mathematics  
The goal of this paper is to prove new recurrences involving 2-colored and 3-colored generalized Frobenius partitions of n similar to the classical recurrence for the partition function p(n ).  ...  The goal of this paper is to prove five new recurrences similar to (1.1) involving 2-colored and 3-colored generalized Frobenius partitions, or F-partitions.  ...  Here we define C~)m(n) as the number of F-partitions of n using m colors [2] . Then C~)m(n) is the number of F-partitions of n using m colors whose order under cyclic permutation of the colors is m.  ... 
doi:10.1016/0012-365x(95)00059-6 fatcat:rz2srbabybcpba7kszcjhh3gsi

Partition Recurrences [article]

Yuriy Choliy, Louis W. Kolitsch, Andrew V. Sills
2018 arXiv   pre-print
We present some Euler-type recurrences for the partition function p(n).  ...  Acknowledgments We thank Dennis Eichhorn and James Sellers for pointing us to several relevant references in the literature, and the anonymous referee who read our manuscript carefully and pointed out  ...  We thank Bruce Landman for organizing the INTEGERS 2016 conference, which facilitated the first and third authors becoming aware of the second author's earlier unpublished work on this topic.  ... 
arXiv:1811.09206v1 fatcat:kj5tbx4qdzevnahmpnnasda3h4

On colored set partitions of type B n

David Wang
2014 Open Mathematics  
Considering the generating function of colored B n-partitions, we get the exact formulas for the expectation and variance of the number of non-zero-blocks in a random colored B n-partition.  ...  We find an asymptotic expression of the total number of colored B n-partitions up to an error of O(n −1/2log7/2 n], and prove that the centralized and normalized number of non-zero-blocks is asymptotic  ...  Acknowledgements The author is grateful to the anonymous referee for the suggestion of generalizing the original -version problem to the present ( )-version problem, and for bringing Dowling's paper to  ... 
doi:10.2478/s11533-014-0419-9 fatcat:dinavoz7tfeadj2ay6mvrfzeu4

Counting Latin rectangles

Ira M. Gessel
1987 Bulletin of the American Mathematical Society  
Explicit formulas for fc = 3 are fairly well known [1-3, 4, pp. 284-286 and 506-507, 5, 6, 9-11, 12, pp. 204-210].  ...  We use properties of the Möbius functions of partition lattices, as did Bogart and Longyear [2], Pranesachar et al. [1, 9] , and Nechvatal [8], but in a somewhat different way.  ...  Explicit formulas for fc = 3 are fairly well known [1] [2] [3] 4 , pp. 284-286 and 506-507, 5, 6, 9-11, 12, pp. 204-210]. Formulas for fc = 4 were found by Pranesachar et al.  ... 
doi:10.1090/s0273-0979-1987-15465-6 fatcat:2m6fsvfo2vhnxbsgmopk7n4jte

Stanley–Elder–Fine theorems for colored partitions [article]

Hartosh Singh Bal, Gaurav Bhatnagar
2021 arXiv   pre-print
More specifically, we give analogous results for b-colored partitions, where each part occurs in b colors; for b-colored partitions with odd parts (or distinct parts); for partitions where the part k comes  ...  in k colors; and, overpartitions.  ...  For example, the partition 4+3+3+2+1+1+1+1 has frequencies: f 1 = 4, f 2 = 1, f 3 = 2, f 4 = 1. One of the quantities in the Stanley-Elder-Fine theorem is for k = 1, 2, . . . , n.  ... 
arXiv:2102.03486v2 fatcat:b5va67p5kjcqjn3cil5z7mojle

Exact and Fixed Parameter Tractable Algorithms for Max-Conflict-Free Coloring in Hypergraphs

Pradeesha Ashok, Aditi Dudeja, Sudeshna Kolay, Saket Saurabh
2018 SIAM Journal on Discrete Mathematics  
In this paper, we initiate a study of a natural maximization version of this problem, namely, Max-CFC: For a given hypergraph H and a fixed r ≥ 2, color the vertices of U using r colors so that the number  ...  For a hypergraph H = (U, F ), a conflict-free coloring of H refers to a vertex coloring where every hyperedge has a vertex with a unique color, distinct from all other vertices in the hyperedge.  ...  Lemmas 16, 17 , and the safeness of the reduction rules 2, 3, 4, 5 together result in the following algorithm for Partitioned p-CFC. Lemma 18.  ... 
doi:10.1137/16m1107462 fatcat:g5ec3lysbfhc5igpndcfdx7kvy

Crossings and nestings in colored set partitions [article]

Eric Marberg
2013 arXiv   pre-print
and 2-Motzkin paths.  ...  Chen, Deng, Du, Stanley, and Yan introduced the notion of k-crossings and k-nestings for set partitions, and proved that the sizes of the largest k-crossings and k-nestings in the partitions of an n-set  ...  Acknowledgements I am grateful to Cyril Banderier, Joel Brewster Lewis, Alejandro Morales, Alexander Postnikov, Steven V Sam, and Richard P. Stanley for helpful discussions and suggestions.  ... 
arXiv:1203.5738v3 fatcat:vt6tub7qvjbuhds3gsiznlbz7a

Detecting Recurrence Domains of Dynamical Systems by Symbolic Dynamics

Peter beim Graben, Axel Hutt
2013 Physical Review Letters  
We propose an algorithm for the detection of recurrence domains of complex dynamical systems from time series.  ...  Our approach exploits the characteristic checkerboard texture of recurrence domains exhibited in recurrence plots (RP).  ...  (e) Phase space partition into recurrence domains for optimal encoding ε * = 1.9. (f) Symbolic recurrence plot [Eq. (2)] of optimal encoding.  ... 
doi:10.1103/physrevlett.110.154101 pmid:25167271 fatcat:vaq4tyav55anznt3bh2wxw7qja

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

Geoffrey Exoo
1994 Electronic Journal of Combinatorics  
For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.  ...  For Ramsey numbers we know R 2 (3) = 6 and R 3 (3) = 17; the current bounds on R 4 (3) are 51 and 65 [5] .  ...  Then define: f 1 (P ) = max{t|P t is sum-free} f 2 (P ) = k i=1 s,t∈Si g n (s, t) and finally f (P ) = c 1 f 1 (P ) + c 2 f 2 (P ).  ... 
doi:10.37236/1188 fatcat:wlzfm7t76ngfle2aq4lpbkqdvu

n-Color partitions with weighted differences equal to minus two

A. K. Agarwal, R. Balasubrananian
1997 International Journal of Mathematics and Mathematical Sciences  
It is shown how these partitions arise in the study of conjugate and self-conjugaten-color partitions. A combinatorial identity for self-conjugaten-color partitions is also obtained.  ...  In this paper we study thosen-color partitions of Agarwal and Andrews, 1987, in which each pair of parts has weighted difference equal to−2Results obtained in this paper for these partitions include several  ...  the conjugate of H and will be denoted by lI For example, if we consider YI 52 + 3, an n-color partition of 8, then l'I 55-2+1 + 33-+ 54 + 33 DEFINITION 2.  ... 
doi:10.1155/s016117129700104x fatcat:sngiow7eczdenhhc3y4w62ngtm

A Computer Proof of a Polynomial Identity Implying a Partition Theorem of Göllnitz

Alexander Berkovich, Axel Riese
2002 Advances in Applied Mathematics  
In addition, we provide computer proofs for new finite analogs of Jacobi and Euler formulas.  ...  In this paper we give a computer proof of a new polynomial identity, which extends a recent result of Alladi and the first author.  ...  Andrews, and Doron Zeilberger for their interest and comments on the manuscript.  ... 
doi:10.1006/aama.2001.0764 fatcat:454eqrs3mzhnrcwus6rt6qubq4

Stirling Numbers of Forests and Cycles

David Galvin, Do Trong Thanh
2013 Electronic Journal of Combinatorics  
Along the way we give recurrences for calculating the generating functions of the sequences $(S(F^c_n,k))_{k \geq 0}$, show that these functions have all real zeroes, and exhibit three different interlacing  ...  For a graph $G$ and a positive integer $k$, the graphical Stirling number $S(G,k)$ is the number of partitions of the vertex set of $G$ into $k$ non-empty independent sets.  ...  Indeed, given a palette of x colors, for each k there are S(G, k) ways to partition the vertex set into k non-empty color classes, and x (k) ways to assign colors the classes.  ... 
doi:10.37236/3170 fatcat:dumts6dakneoxjrsvj4jiswkfm

Stirling numbers of forests and cycles [article]

Do Trong Thanh, David Galvin
2012 arXiv   pre-print
Along the way we give recurrences for calculating the generating functions of the sequences (S(F^c_n,k))_k ≥ 0, show that these functions have all real zeroes, and exhibit three different interlacing patterns  ...  For a graph G and a positive integer k, the graphical Stirling number S(G,k) is the number of partitions of the vertex set of G into k non-empty independent sets.  ...  Indeed, given a palette of x colors, for each k there are S(G, k) ways to partition the vertex set into k non-empty color classes, and x (k) ways to assign colors the classes.  ... 
arXiv:1206.3591v1 fatcat:xyyrufeojbab7frt2glb2g36xq

Chunking as a rational strategy for lossy data compression in visual working memory

Matthew R. Nassar, Julie C. Helmers, Michael J. Frank
2018 Psychological review  
We show that such chunking can: 1) facilitate performance 7 improvements for abstract capacity-limited systems, 2) be optimized through 8 reinforcement, 3) be implemented by center-surround dynamics, and  ...  The nature of capacity limits for visual working memory has been the 3 subject of an intense debate that has relied on models that assume items are 4 encoded independently.  ...  Capacity limitations are modeled by a fixed limit on the 2 length of the resulting "sentence" comprised of color and orientation words separated by word 3 termination symbols (2/3 for color/orientation  ... 
doi:10.1037/rev0000101 pmid:29952621 pmcid:PMC6026019 fatcat:ztijqj33wzbyrkttihr3pxwgdi

Probabilistic analysis of algorithms for the Dutch national flag problem

Wei-Mei Chen
2005 Theoretical Computer Science  
A detailed probabilistic analysis is given of algorithms for the Dutch national flag problem. We derive central and local limit theorems for the cost, as well as probabilities of large deviations.  ...  Then F n (y) satisfies the recurrence F n (y) = y n−2 j =0 n − j − 1 2 n−j F j (y) + n + 1 2 n (n 2), (1) with F 0 (y) = F 1 (y) = 1. Proof.  ...  Then H n (y) satisfies the recurrence H n (y) = 1 3 n + 1 + y 3 H n−1 (y) + (y 2 + y) n−2 j =0 H j (y) 3 n−j (n 2), (3) with H 0 (y) = 1 and H 1 (y) = (2 + y)/3. Proof.  ... 
doi:10.1016/j.tcs.2005.03.047 fatcat:53cyivliqjetxjnpu4d2orsi2i
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