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Recurrences for 2-colored and 3-colored F-partitions
1996
Discrete Mathematics
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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]
2018
arXiv
pre-print
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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INTEGERS 2016 conference, which facilitated the first and third authors becoming aware of the second author's earlier unpublished work on this topic.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
2014
Open Mathematics
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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
1987
Bulletin of the American Mathematical Society
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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]
2021
arXiv
pre-print
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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
2018
SIAM Journal on Discrete Mathematics
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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]
2013
arXiv
pre-print
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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
2013
Physical Review Letters
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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
1994
Electronic Journal of Combinatorics
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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
1997
International Journal of Mathematics and Mathematical Sciences
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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
2002
Advances in Applied Mathematics
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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
2013
Electronic Journal of Combinatorics
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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]
2012
arXiv
pre-print
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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
2018
Psychological review
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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
2005
Theoretical Computer Science
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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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