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Totally Free Expansions of Matroids

James Geelen, James Oxley, Dirk Vertigan, Geoff Whittle
2002 Journal of combinatorial theory. Series B (Print)  
, over all totally free minors N of M, of the number of inequivalent F-representations of N.  ...  A 3-connected matroid N is totally free if neither N nor its dual has a fixed element whose deletion is a series extension of a 3-connected matroid.  ...  ACKNOWLEDGMENTS Geelen was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada, Oxley and Vertigan were partially supported by grants from the National Security  ... 
doi:10.1006/jctb.2001.2068 fatcat:3qtpgglihrgdrepvnug3h2uqqu

A note on clone sets in representable matroids

Talmage J. Reid, Jakayla R. Robbins
2006 Discrete Mathematics  
We investigate the size of clone sets in representable matroids. Theorem 1.1. A 2-connected binary matroid has no clone sets.  ...  If M is a 3-connected GF(3)-representable matroid that contains a clone set, then MU 2,4 .  ...  A matroid M is totally free if and only if it is a totally free expansion of U 2,4 . Lemma 2.3 (Geelen et al. [4, Proposition 4.3]).  ... 
doi:10.1016/j.disc.2006.05.005 fatcat:c33rfthhnvdydfvxl5ypl654qy

Page 7039 of Mathematical Reviews Vol. , Issue 2002J [page]

2002 Mathematical Reviews  
free expansions of matroids.  ...  (iii) The identification of all totally free quaternary and quinternary matroids.  ... 

On clone sets of GF(q)-representable matroids

Talmage J. Reid, Xiangqian Zhou
2009 Discrete Mathematics  
This bound is given by investigating the representability of a class of near-uniform matroids.  ...  We bound the size of a clone set in a 3-connected non-uniform GF(q)-representable matroid by a linear function of q.  ...  We thank the anonymous referee for comments which improved the proofs and results of Section 4.  ... 
doi:10.1016/j.disc.2008.02.032 fatcat:rcjb6zfzsvgtpnkvo2luewvxla

Revenue submodularity

Shaddin Dughmi, Tim Roughgarden, Mukund Sundararajan
2009 Proceedings of the tenth ACM conference on Electronic commerce - EC '09  
We also give two applications of revenue submodularity: good approximation algorithms for novel market expansion problems, and approximate revenue guarantees for the VCG mechanism with i.i.d. bidders.  ...  in matroid markets with i.i.d. bidders and "sufficient competition".  ...  Thus, as in the proof of Theorem 3.2, the VCG mechanism inadvertently maximizes the total virtual value over the bases of the matroid.  ... 
doi:10.1145/1566374.1566409 dblp:conf/sigecom/DughmiRS09 fatcat:3k4icatlt5gq5cueaurizp7hd4

Revenue Submodularity [chapter]

Shaddin Dughmi, Tim Roughgarden, Mukund Sundararajan
2009 Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering  
We also give two applications of revenue submodularity: good approximation algorithms for novel market expansion problems, and approximate revenue guarantees for the VCG mechanism with i.i.d. bidders.  ...  in matroid markets with i.i.d. bidders and "sufficient competition".  ...  Thus, as in the proof of Theorem 3.2, the VCG mechanism inadvertently maximizes the total virtual value over the bases of the matroid.  ... 
doi:10.1007/978-3-642-03821-1_13 fatcat:yur37nu7ezfvpjyag4vrrzrcpu

Revenue Submodularity

Shaddin Dughmi, Tim Roughgarden, Mukund Sundararajan
2012 Theory of Computing  
We also give two applications of revenue submodularity: good approximation algorithms for novel market expansion problems, and approximate revenue guarantees for the VCG mechanism with i.i.d. bidders.  ...  in matroid markets with i.i.d. bidders and "sufficient competition."  ...  Thus, as in the proof of Theorem 3.2, the VCG mechanism inadvertently maximizes the total virtual value over the bases of the matroid.  ... 
doi:10.4086/toc.2012.v008a005 dblp:journals/toc/DughmiRS12 fatcat:hahb7ynpxrc3xljqmatya6ak2u

Network theory and transversal matroids

András Recski, Masao Iri
1980 Discrete Applied Mathematics  
Finally, we consider this classification of the networks and give matroidal characterization of the hierarchy classes. 311 312 A. Recski, M. Id  ...  Various matroidal models for the solvability of active linear networks are described in a uni&d way and compared with one another.  ...  in the expansion of det N.  ... 
doi:10.1016/0166-218x(80)90027-x fatcat:olfmcau25fawxgrku63ife4mry

A quasisymmetric function for matroids

Louis J. Billera, Ning Jia, Victor Reiner
2009 European journal of combinatorics (Print)  
A new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant:  ...  minimize the total weight; • behaves simply under matroid duality; • has a simple expansion in terms of P-partition enumerators; • is a valuation on decompositions of matroid base polytopes.  ...  of a free commutative algebra, that is, a polynomial algebra.  ... 
doi:10.1016/j.ejc.2008.12.007 fatcat:2nnu5o6mafafbbxxiszazzxpsa

A quasisymmetric function for matroids [article]

Louis J. Billera, Ning Jia, Victor Reiner
2020 arXiv   pre-print
of how many bases minimize the total weight, (4) behaves simply under matroid duality, (5) has a simple expansion in terms of P-partition enumerators, and (6) is a valuation on decompositions of matroid  ...  for integer weight vectors that give minimum total weight to a unique base of the matroid, (3) is equivalent, via the Hopf antipode, to a generating function for integer weight vectors which keeps track  ...  Let Mat be the free Z-module consisting of formal Z-linear combinations of basis elements [M ] indexed by isomorphism classes of matroids M .  ... 
arXiv:math/0606646v4 fatcat:3pvkdhqkzfbm5ny57jxb7ydzae

Page 55 of Mathematical Reviews Vol. , Issue 92a [page]

1992 Mathematical Reviews  
In the authors’ terminology, a symmetric function is said to be multiplicity free if its Schur function expansion has all coefficients equal to 0, 1 or —1.  ...  exists a k-term arithmetic progression A which totally belongs to one of the A;’s.  ... 

Page 7306 of Mathematical Reviews Vol. , Issue 2003j [page]

2003 Mathematical Reviews  
This result follows from 05 COMBINATORICS 7306 an expansion of p(A; A) in terms of the free convex sets of an an- timatroid A [G. Gordon and E. W.  ...  The authors study this anti- matroid characteristic polynomial for the class of chordal graphs. A graph is chordal if every cycle of length 4 or greater has a chord.  ... 

Ported Tutte Functions of Extensors and Oriented Matroids [article]

Seth Chaiken
2006 arXiv   pre-print
The corank-nullity polynomial, basis expansions with activities, and a geometric lattice expansion generalize to ported Tutte functions of oriented matroids.  ...  Solutions F can distinguish different orientations of the same matroid.  ...  Ariel Caticha, Physics Department of the University at Albany, for his invitation to participate in his graduate Geometry of Physics course; Joanna Ellis-Monaghan, Michel Las Vergnas, Gary Gordon, Alan  ... 
arXiv:math/0605707v2 fatcat:th3pufbranahhlo73y74f3giaa

Investigating solvability and complexity of linear active networks by means of matroids

B. Petersen
1979 IEEE Transactions on Circuits and Systems  
I/, in case of regularity is the free mutroid, i.e., M-(E$u 1,') contains no circuits.  ...  Now a subset of columns of B' is linearly independent iff the columns correspond to a cutset-free subgraph of H [20] .  ...  Professor at the Department of Mathematics, University of Denmark Lyngby. He has carried out research in the fields of circuit theory, graph theory, matroid theory, and computational complexity.  ... 
doi:10.1109/tcs.1979.1084645 fatcat:7gbkyyeshbfldiipdk6fcwpq24

The image of a tropical linear space [article]

Joshua Mundinger
2018 arXiv   pre-print
We show under mild hypotheses that tropim_A(L) is realizable if L is and apply the tropical image to construct the stable sum of two tropical linear spaces without a disjoint pair of bases.  ...  Given a tropical linear space L ⊆T^n and a matrix A ∈T^m × n, the image AL of L under A is typically not a tropical linear space.  ...  Transversal matroids are the induced matroids of free matroids. Corollary 4.7 also follows from the description of Stiefel tropical linear spaces as the tropical image of free modules.  ... 
arXiv:1808.02150v1 fatcat:f5p56cripfaqvirgazt5c5lnd4
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