Balanced Incomplete Block Designs and t-Designs2-(v,k,l) Designs of Small OrderBIBDs with Small Block Sizet-Designs, t = 3Steiner SystemsSymmetric DesignsResolvable and Near Resolvable DesignsLatin Squares, MOLS, and Orthogonal ArraysLatin SquaresMutually Orthogonal Latin Squares (MOLS)Incomplete MOLSOrthogonal Arrays of Index More Than OneOrthogonal Arrays of Strength More Than TwoPairwise Balanced DesignsPBDs and GDDs: The BasicsPBDs: Recursive ConstructionsPBD-ClosurePairwise Balanced Designs as Linear SpacesPBDs and GDDs of Higher IndexPBDs, Frames, and ResolvabilityOther Combinatorial DesignsAssociation SchemesBalanced (Part) Ternary DesignsBalanced Tournament DesignsBhaskar Rao DesignsComplete Mappings and Sequencings of Finite GroupsConfigurationsCostas ArraysCoveringsCycle SystemsDifference FamiliesDifference MatricesDifference Sets: AbelianDifference Sets: NonabelianDifference Triangle SetsDirected DesignsD-Optimal MatricesEmbedding Partial QuasigroupsEquidistant Permutation ArraysFactorial DesignsFrequency SquaresGeneralized QuadranglesGraph Decompositions and DesignsGraphical DesignsHadamard Matrices and DesignsHall Triple SystemsHowell DesignsMaximal Sets of MOLSMendelsohn DesignsThe Oberwolfach ProblemOrdered Designs and Perpendicular ArraysOrthogonal DesignsOrthogonal Main Effect PlansPackingsPartial GeometriesPartially Balanced Incomplete Block DesignsQuasigroupsQuasi-Symmetric Designs(r,l)-DesignsRoom SquaresSelf-Orthogonal Latin Squares (SOLS)SOLS with a Symmetric Orthogonal Mate (SOLSSOM)Sequences with Zero AutocorrelationSkolem SequencesSpherical t-DesignsStartersTrades and Defining Sets(t,m,s)-NetsTuscan Squarest-Wise Balanced DesignsUniformly Resolvable DesignsVector Space DesignsWeighing Matrices and Conference MatricesWhist TournamentsYouden Designs, GeneralizedYouden SquaresApplicationsCodesComputer Science: Selected ApplicationsApplications of Designs to CryptographyDerandomizationOptimality and Efficiency: Comparing Block DesignsGroup TestingScheduling a TournamentWinning the LotteryRelated Mathematics and Computational MethodsFinite Groups and DesignsNumber Theory and Finite FieldsGraphs and MultigraphsFactorizations of GraphsStrongly Regular GraphsTwo-GraphsClassical GeometriesProjective Planes, NondesarguesianComputational Methods in Design TheoryIndex

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CRC Handbook of Combinatorial Designs

Decomposition of regular matroids

This is a direct continuation of Shellable Nonpure Complexes and Posets. I, which appeared in Transactions of the American Mathematical Society 348 (1996), 1299-1327. 8. Interval-generated lattices and dominance order In this section and the following one we will continue exemplifying the applicability of lexicographic shellability to nonpure posets. Let F = {I1, I2, . . . , In} be a family of intervals of integers, by which is meant sets of the form [a, b] = {a, a + 1, . . . , b}, a ≤ b. We assume that there are no containments among these intervals, and that they are ordered so that their left and right endpoints are increasing. Let L(F) be the lattice of all sets that are unions of subfamilies of F , ordered by inclusion. Such interval-generated lattices L(F) were introduced and studied by Greene [G]. Define an edge-labeling λ of L(F) as follows. If A → B is a covering and a = max(B \A), then λ(A→ B) = { −a, if (a+ 1) ∈ A and a is the left endpoint of some I ∈ F ,

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Shellable nonpure complexes and posets. II

We show that determining the Jones polynomial of an alternating link is #P-hard. This is a special case of a wide range of results on the general intractability of the evaluation of the Tutte polynomial T(M; x, y) of a matroid M except for a few listed special points and curves of the (x, y)-plane. In particular the problem of evaluating the Tutte polynomial of a graph at a point in the (x, y)-plane is #P-hard except when (x − 1)(y − 1) = 1 or when (x, y) equals (1, 1), (−1, −1), (0, −1), (−1, 0), (i, −i), (−i, i), (j, j2), (j2, j) where j = e2πi/3

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0305004100068936

On the computational complexity of the Jones and Tutte polynomials

The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have been already evaluated, together with explanations which tell in which contexts they have appeared. Third, it points out references where further such determinant evaluations can be found.

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Advanced determinant calculus

Matroid Applications: The Tutte Polynomial and Its Applications

A construction based on work by Tutte and Grothendieck is applied to a decomposition on combinatorial pregeometries in order to study an important class of invariants. The properties of this Tutte decomposition of a pregeometry into a subgeometry G\e and contraction G/e is explored in a categorically integrated view using factored strong maps. After showing that direct sum decomposition distributes over the Tutte decomposition we construct a universal pair (R, t) where R is a free commutative ring with two generators corresponding to a loop and an isthmus; and t, the Tutte polynomial assigns a ring element to each pregeometry. Evaluations of t(G) give the Mobius function, characteristic polynomial, Crapo invariant, and numbers of subsets, bases, spanning and independent sets of G and its Whitney dual. For geometries a similar decomposition gives the same information as the chromatic polynomial throwing new light on the critical problem. A basis is found for all linear identities involving Tutte polynomial coefficients. In certain cases including Hartmanis partitions one can recover all the Whitney numbers of the associated geomnetric lattice L(G) from t(G) and conversely. Examples and counterexamples show that duals, minors, connected pregeometries, series-parallel networks, free geometries (on which many invariants achieve their upper bounds), and lower distributive pregeometries are all characterized by their polynomials. However, inequivalence, Whitney numbers, and representability are not always invariant. Applying the decomposition to chain groups we generalize the classical twocolor theorem for graphs to show when a geometry can be imbedded in binary affine space. The decomposition proves useful also for graphical pregeometries and for unimodular (orientable) pregeometries in the counting of cycles and coboundaries.

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A decomposition for combinatorial geometries

The category of pregeometries with basepoint is defined and explored. In this category two important operations are extensively characterized: the series connection S(G, H), and the parallel connection P(G, H) = S(G, H); and the latter is shown to be the categorical direct sum. For graphical pregeometries, these notions coincide with the classical definitions. A pregeometry Fis a nontrivial series (or parallel) connection relative to a basepoint p iff the deletion F\\p (contraction F/p) is separable. Thus both connections are zz-ary symmetric operators with identities and generate a free algebra. Elements of the subalgebra A[C2l generated by the two point circuit are defined as series-parallel networks, and this subalgebra is shown to be closed under arbitrary minors. Nonpointed series-parallel networks are characterized by a number of equivalent conditions: 1. They are in A[C2] relative to some point. 2. They are in A[C2] relative to any point. For any connected minor K of three or more points: 3. K is not the four point line or the lattice of partitions of a four element set. 4. K or K is not a geometry. 5. For any point e in K, K\\e or K/e is separable. Series-parallel networks can also be characterized in a universally constructed ring of pregeometries generalized from previous work of W. Tutte and A. Grothendieck. In this Tutte-Grothendieck ring they are the pregeometries for which the Crapo invariant equals one. Several geometric invariants are directly calculated in this ring including the complexity and the chromatic polynomial. The latter gives algebraic proofs of the two and three color theorems.

https://www.ams.org/tran/1971-154-00/S0002-9947-1971-0288039-7/S0002-9947-1971-0288039-7.pdf

A combinatorial model for series-parallel networks

The lattice of integer partitions

R. Stanley, in an investigation of modular flats in geometries (Algebra Universalis 1-2 (1971), 214-217), proved that the characteristic polynomial x(x) of a modular flat x divides the characteristic polynomial x(G) of a geometry G. In this paper we identify the quotient: THEOREM. If x is a modular flat of G, x(G)/x(x) = x(Tx(G))I(X 1), where Tl(G) is the complete Brown truncation of G by x. (The lattice of TX(G) consists of all flats containing x and all flats disjoint from x, with the induced order from G.) We give many characterizations of modular flats in terms of their lattice properties as well as by means of a short-circuit axiom and a modular version of the MacLane-Steinitz exchange axiom. Modular flats are shown to have many of the useful properties of points and distributive flats (separators) in addition to being much more prevalent. The theorem relating the chromatic polynomials of two graphs and the polynomial of their vertex join across a common clique generalizes to geometries: THEOREM. Given geometries G and H, if x is a modular flat of G as well as a subgeometry of H, then there exists a geometry P = PX(G, H) which is a pushout in the category of injective strong maps and such that X(P) = x(G)X(H)/x(x). The closed set structure, rank function, independent sets, and lattice properties of P are characterized. After proving a modular extension theorem we give applications of our results to Crapo's single element extension theorem, Crapo's join operation, chain groups, unimodular geometries, transversal geometries, and graphs.

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Tom Brylawski

Papers

Matroid Applications: The Tutte Polynomial and Its Applications

A decomposition for combinatorial geometries

A combinatorial model for series-parallel networks

The lattice of integer partitions

Modular constructions for combinatorial geometries