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, 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
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
free expansions of matroids. ... (iii) The identification of all totally free quaternary and quinternary matroids. ...
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
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
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
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
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 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
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
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. ...
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. ...
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
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  . ... 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
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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