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A lattice path approach to counting partitions with minimum rank t
2002
Discrete Mathematics
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In this paper, we give a combinatorial proof via lattice paths of the following result due to Andrews and Bressoud: for t 6 1, the number of partitions of n with all successive ranks at least t is equal ...
to the number of partitions of n with no part of size 2 − t. ...
Introduction In this paper, we show how to use a lattice path counting technique to establish a relationship between partitions deÿned by rank conditions and partitions with forbidden part sizes. ...
doi:10.1016/s0012-365x(01)00225-4
fatcat:2ue7oddkljgn5hbkvtzbl5df6m
Metrics for Clustering Comparison in Bioinformatics
2016
Proceedings of the 5th International Conference on Pattern Recognition Applications and Methods
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This yields an exact (rather than heuristic) approach to the consensus partition (combinatorial optimization) problem. ...
From both theoretical and applicative perspectives, a useful and interesting distance between any two partitions is HD, which counts the number of atoms finer than either one but not both. ...
The approach allows for generalizations in that the size may be replaced with any alternative orderpreserving lattice function, such as the rank (or the entropy of partitions). ...
doi:10.5220/0005707102990308
dblp:conf/icpram/Rossi16a
fatcat:6oxbxnrkhjf4nivlewzbxn7ava
Diamond-colored distributive lattices, move-minimizing games, and fundamental Weyl symmetric functions: The type A case
[article]
2017
arXiv
pre-print
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The objective of this game is to find, if possible, the least number of "domino moves" to get from one partition to another, where a domino move is, with one exception, the addition or removal of a domino-shaped ...
to as the "type A fundamental lattices." ...
So, since the rank of a partition is simply the minimum number of moves required to go from the minimum partition to the given partition, the rank of σ is given by ρ(σ) = c i . Example 5.12. ...
arXiv:1702.00806v1
fatcat:uc6aldt4kbg6pfjzxa5qompuia
Unimodality and Dyck paths
[article]
2012
arXiv
pre-print
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We propose an original approach to the problem of rankunimodality for Dyck lattices. ...
It is based on a well known recursive construction of Dyck paths originally developed in the context of the ECO methodology, which provides a partition of Dyck lattices into saturated chains. ...
Indeed, this is a special case of the unimodality of Gaussian coefficients (counting integer partitions fitting inside a rectangle with respect to their size), which has been proved in several sources ...
arXiv:1207.7295v1
fatcat:7kun5vmlv5gyfb6evmzv3mm2lu
A note on rank complement of rational Dyck paths and conjugation of $(m,n)$-core partitions
2017
Journal of Combinatorics
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go below the diagonal; (m, n)-cores are partitions with no hook length equal to m or n. ...
This leads to: i) a new approach to characterizing n-cores; ii) a simple approach for counting the number of self-conjugate (m, n)-cores; iii) a proof of the equivalence of two conjectured combinatorial ...
We claim that b < n, so that h is the rank of a lattice If a partition λ is an (m, n) core, then clearly so is its conjugate λ T . ...
doi:10.4310/joc.2017.v8.n4.a7
fatcat:kseq55gjuvhdfdv7cynlyfke4i
AlphaSum
2009
Proceedings of the 12th International Conference on Extending Database Technology Advances in Database Technology - EDBT '09
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To tackle with the inherent complexity, we rely on the properties of the fuzzy interpretation to further develop a novel ranked set cover based evaluation mechanism (RSC). ...
We formulate the problem of table summarization with the help of value lattices. ...
Given a weighted value clustering lattice L, a lattice node vj is a clustering of a lattice node vi, denoted by vi vj, if ∃path p = vi vj in L. We also say that vj covers vi. ...
doi:10.1145/1516360.1516373
dblp:conf/edbt/CandanCQS09
fatcat:j245eucyijc2letlapyeq6jjmi
Top-k-size keyword search on tree structured data
2015
Information Systems
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To address performance challenges our algorithms exploit a lattice of the partitions of the keyword set, which empowers a linear time performance. ...
Recent keyword search approaches return lowest common ancestors (LCAs) of the keyword matches ranked with respect to their relevance to the keyword query. ...
In our context, every path from the source partition to the sink partition represents a unique way of combining keyword instances to partial LCAs to full LCAs. ...
doi:10.1016/j.is.2014.07.002
fatcat:jvocpjzbqrdi7e5mxmwike2c5e
Rank complement of rational Dyck paths and conjugation of (m,n)-core partitions
[article]
2015
arXiv
pre-print
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the diagonal; (m,n)-cores are partitions with no hook length equal to m or n.Anderson established a bijection between (m,n)-Dyck paths and (m,n)-cores. ...
This leads to: i) a new approach to characterizing n-cores; ii) a simple approach for counting the number of self-conjugate (m,n)-cores; iii) a proof of the equivalence of two conjectured combinatorial ...
The author is very grateful to Professor Adriano Garsia for inspirations and encouraging conversations. ...
arXiv:1504.02075v2
fatcat:f4syi5vyxvarfcdefjs6b7bxvm
Noncrossing partitions
2000
Discrete Mathematics
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Without attempting to give an exhaustive account, this article illustrates aspects of enumerative and algebraic combinatorics supported by the lattice of noncrossing partitions, re ecting developments ...
In a 1972 paper, Germain Kreweras investigated noncrossing partitions under the reÿnement order relation. ...
to her attention valuable references. ...
doi:10.1016/s0012-365x(99)00273-3
fatcat:6qht3p2ygzcypgniitb3wyiveq
Weighted paths between partitions
[article]
2016
arXiv
pre-print
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With one exception, all of them obtain as minimum-weight paths in the undirected graph corresponding to the Hasse diagram of the partition lattice. ...
Secondly, another partition distance known as "variation of information" is seen to correspond to a minimum-weight path with edge weights determined by the entropy of partitions. ...
The geometric approach adopted here enables to analyze further partition distances obtained by replacing the size with alternative partition functions such as entropy, rank, logical entropy and co-size ...
arXiv:1509.01852v2
fatcat:iroivm54jndhnfmotakz2ojjpq
Fourier meets Möbius: fast subset convolution
[article]
2006
arXiv
pre-print
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To demonstrate the applicability of fast subset convolution, we present the first O^*(2^k n^2 + n m) algorithm for the minimum Steiner tree problem in graphs with n vertices, k terminals, and m edges with ...
We also discuss extensions to recent O^*(2^n)-time algorithms for covering and partitioning problems (Björklund and Husfeldt, FOCS 2006; Koivisto, FOCS 2006). ...
A path joins x 1 to x ℓ+1 . A hypergraph is connected if for all distinct x, y ∈ V there exists a path joining x to y. ...
arXiv:cs/0611101v1
fatcat:zvrm7bgjwbew7mw5pocjriqsku
Author index
2002
Discrete Mathematics
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, A., S. Corteel, A. Postnikov and C. D. Savage, A lattice path approach to counting partitions with minimum rank t (1-3) 31-39 Chen, G. and X. Yu, A note on fragile graphs (1-3) 41-43 Chen, G., A. ...
Schelp, Two-edge colorings of graphs with bounded degree in both , C.W. and D.B. West, A Fibonacci tiling of the plane (1-3) 111-116 Jacobson, M.S., see R.J. ...
doi:10.1016/s0012-365x(02)00277-7
fatcat:s5wqhobycrgn3kc2st32twlh4m
Page 3198 of Mathematical Reviews Vol. , Issue 2003e
[page]
2003
Mathematical Reviews
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E.] (1-CA; Berkeley, CA); Savage, Carla D. (1-NCS-C; Raleigh, NC) A lattice path approach to counting partitions with minimum rank /. ...
Summary: “In this paper, we give a combinatorial proof via lattice paths of the following result due to Andrews and Bressoud: for t <1, the number of partitions of ” with all successive ranks at least ...
Fourier meets möbius: fast subset convolution
2007
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing - STOC '07
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We present a fast algorithm for the subset convolution problem: given functions f and g defined on the lattice of subsets of an n-element set N , compute their subset convolution f * g, defined for all ...
A path joins x1 to x ℓ+1 . A hypergraph is connected if for all distinct x, y ∈ V there exists a path joining x to y. ...
For example, considering a slightly different convolution operation of the form h ′ (S) = X T ⊆N f`T´g`S∆T´, (2) where S∆T = (S \ T ) ∪ (T \ S), the Fourier approach immediately yields an evaluation approach ...
doi:10.1145/1250790.1250801
dblp:conf/stoc/BjorklundHKK07
fatcat:hvtaeoesqfdo5hnekpxequsb5e
A Natural Family of Flag Matroids
2007
SIAM Journal on Discrete Mathematics
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These flag matroids can also be defined in terms of lattice paths and they provide a generalization of the lattice path matroids of [Bonin et al., JCTA 104 (2003)]. ...
A flag matroid can be viewed as a chain of matroids linked by quotients. ...
Acknowledgements The author thanks Dominic Welsh for asking a question that suggested this work, and Joseph Bonin and Sergi Elizalde for helpful discussions. ...
doi:10.1137/050627873
fatcat:ss7gsbtm5jewnij7xnvjvr76em
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