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Totally Free Expansions of Matroids

2002
*
Journal of combinatorial theory. Series B (Print)
*

, over all

doi:10.1006/jctb.2001.2068
fatcat:3qtpgglihrgdrepvnug3h2uqqu
*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 ...##
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A note on clone sets in representable matroids

2006
*
Discrete Mathematics
*

We investigate the size

doi:10.1016/j.disc.2006.05.005
fatcat:c33rfthhnvdydfvxl5ypl654qy
*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]). ...##
###
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*. ...

##
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On clone sets of GF(q)-representable matroids

2009
*
Discrete Mathematics
*

This bound is given by investigating the representability

doi:10.1016/j.disc.2008.02.032
fatcat:rcjb6zfzsvgtpnkvo2luewvxla
*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. ...##
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Revenue submodularity

2009
*
Proceedings of the tenth ACM conference on Electronic commerce - EC '09
*

We also give two applications

doi:10.1145/1566374.1566409
dblp:conf/sigecom/DughmiRS09
fatcat:3k4icatlt5gq5cueaurizp7hd4
*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*. ...##
###
Revenue Submodularity
[chapter]

2009
*
Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering
*

We also give two applications

doi:10.1007/978-3-642-03821-1_13
fatcat:yur37nu7ezfvpjyag4vrrzrcpu
*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*. ...##
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Revenue Submodularity

2012
*
Theory of Computing
*

We also give two applications

doi:10.4086/toc.2012.v008a005
dblp:journals/toc/DughmiRS12
fatcat:hahb7ynpxrc3xljqmatya6ak2u
*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*. ...##
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Network theory and transversal matroids

1980
*
Discrete Applied Mathematics
*

Finally, we consider this classification

doi:10.1016/0166-218x(80)90027-x
fatcat:olfmcau25fawxgrku63ife4mry
*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. ...##
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A quasisymmetric function for matroids

2009
*
European journal of combinatorics (Print)
*

A new isomorphism invariant

doi:10.1016/j.ejc.2008.12.007
fatcat:2nnu5o6mafafbbxxiszazzxpsa
*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. ...##
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A quasisymmetric function for matroids
[article]

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 . ...

##
###
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. ...##
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Ported Tutte Functions of Extensors and Oriented Matroids
[article]

2006
*
arXiv
*
pre-print

The corank-nullity polynomial, basis

arXiv:math/0605707v2
fatcat:th3pufbranahhlo73y74f3giaa
*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 ...##
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Investigating solvability and complexity of linear active networks by means of matroids

1979
*
IEEE Transactions on Circuits and Systems
*

I/, in case

doi:10.1109/tcs.1979.1084645
fatcat:7gbkyyeshbfldiipdk6fcwpq24
*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. ...##
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The image of a tropical linear space
[article]

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

arXiv:1808.02150v1
fatcat:f5p56cripfaqvirgazt5c5lnd4
*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. ...
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