A filtration of -Catalan numbers

The operator of F. Bergeron, Garsia, Haiman and Tesler [F. Bergeron, A. Garsia, M. Haiman, G. Tesler, Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions, Methods Appl. Anal. 6 (1999) 363–420] acting on the k-Schur functions [L. Lapointe, A. Lascoux, J. Morse, Tableaux atoms and a new Macdonald positivity conjecture, Duke Math. J. 116 (2003) 103–146; L. Lapointe, J. Morse, Schur functions analogs for a filtration of the symmetric functions space, J. Combin. Theory Ser. A 101 (2003) 191–224; L. Lapointe, J. Morse, Tableaux on k+1-cores, reduced words for affine permutations and k-Schur expansion, J. Combin. Theory Ser. A 112 (2005) 44–81] indexed by a single column has a coefficient in the expansion which is an analogue of the (q,t)-Catalan number with a level k. When k divides n we conjecture a representation theoretical model in this case such that the graded dimensions of the module are the coefficients of the (q,t)-Catalan polynomials of level k. When the parameter t is set to 1, the Catalan numbers of level k are shown to count the number of Dyck paths that lie below a certain Dyck path with q counting the area of the path.     

Sporadic groups and automorphisms of linear spaces

This article is a contribution to the study of the automorphism groups of finite linear spaces. In particular we look at almost simple groups and prove the following theorem: Let G be an almost simple group and let 𝒮 be a finite linear space on which G acts as a line‐transitive automorphism group. Then the socle of G is not a sporadic group. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 353–362, 2000     

A New Monohedral Hexamorphic Prototile

In this paper we apply the fitting method and a result due to Martin (see [“Polynominoes—A Guide to Puzzles and Problems in Tiling,” The Mathematical Association of American, New York, 1991] to get a hexamorphic prototile different from the only one we have known, presented by Fontaine and Martin in [J. Combin. Theory Ser. A34(1983), 119–121].     

A classification of tightly attached half-arc-transitive graphs of valency 4

A graph is said to be half-arc-transitive if its automorphism group acts transitively on the set of its vertices and edges but not on the set of its arcs. With each half-arc-transitive graph of valency 4 a collection of the so-called alternating cycles is associated, all of which have the same even length. Half of this length is called the radius of the graph in question. Moreover, any two adjacent alternating cycles have the same number of common vertices. If this number, the so-called attachment number, coincides with the radius, we say that the graph is tightly attached. In [D. Marušič, Half-transitive group actions on finite graphs of valency 4, J. Combin. Theory Ser. B 73 (1998) 41–76], Marušič gave a classification of tightly attached half-arc-transitive graphs of valency 4 with odd radius. In this paper the even radius tightly attached graphs of valency 4 are classified, thus completing the classification of all tightly attached half-arc-transitive graphs of valency 4.     

Extending the Greene-Kleitman theorem to directed graphs

The celebrated Dilworth theorem (Ann. of Math. 51 (1950), 161–166) on the decomposition of finite posets was extended by Greene and Kleitman (J. Combin. Theory Ser. A 20 (1976), 41–68). Using the Gallai-Milgram theorem (Acta Sci. Math. 21 (1960), 181–186) we prove a theorem on acyclic digraphs which contains the Greene-Kleitman theorem. The method of proof is derived from M. Saks' elegant proof (Adv. in Math. 33 (1979), 207–211) of the Greene-Kleitman theorem.     

Pooling spaces associated with finite geometry

Motivated by the works of Ngo and Du [H. Ngo, D. Du, A survey on combinatorial group testing algorithms with applications to DNA library screening, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 55 (2000) 171–182], the notion of pooling spaces was introduced [T. Huang, C. Weng, Pooling spaces and non-adaptive pooling designs, Discrete Mathematics 282 (2004) 163–169] for a systematic way of constructing pooling designs; note that geometric lattices are among pooling spaces. This paper attempts to draw possible connections from finite geometry and distance regular graphs to pooling spaces: including the projective spaces, the affine spaces, the attenuated spaces, and a few families of geometric lattices associated with the orbits of subspaces under finite classical groups, and associated with d-bounded distance-regular graphs.     

Tactical decompositions and some configurationsv 4

The study of configurations or — more generally — finite incidence geometries is best accomplished by taking into account also their automorphism groups. These groups act on the geometry and in particular on points, blocks, flags and even anti-flags. The orbits of these groups lead to tactical decompositions of the incidence matrices of the geometries or of related geometries. We describe the general procedure and use these decompositions to study symmetric configurationsv ₄ for smallv. Tactical decompositions have also been used to construct all linear spaces on 12 points [2] and all proper linear spaces on 17 points [3].     

Counting 1-factors in infinite graphs

F. Bry (J. Combin. Theory Ser. B 34 (1983), 48–57) proved that a locally finite infinite n-connected factorizable graph has at least (n−1)! 1-factors and showed that for n = 2 this lower bound is sharp. We prove that for n≥3 any infinite n-connected factorizable graph has at least n! 1-factors (which is a sharp lower bound).     

Hadamard matrices of order 28 with automorphisms of order 7

The only primes which can divide the order of the automorphism group of a Hadamard matrix of order 28 are 13, 7, 3, and 2, and there are precisely four inequivalent matrices with automorphisms of order 13 (Tonchev, J. Combin. Theory Ser. A35 (1983), 43–57). In this paper we show that there are exactly twelve inequivalent Hadamard matrices of order 28 with automorphisms of order 7. In particular, there are precisely seven matrices with transitive automorphism groups.     

A new lower bound on the strong connectivity of an oriented graph. Application to diameters with a particular case related to Caccetta Häggkvist conjecture

In this paper, by using minimum out-degree and minimum in-degree, we give a new lower bound on the vertex-strong connectivity of an oriented graph. In the case of a tournament, our lower bound improves that of Thomassen obtained in 1980 and which use the notion of irregularity (see [C. Thomassen, Hamiltonian-connected tournaments, J. Combin. Theory Ser. B 28 (1980) 142–163]). As application, we determine a pertinent upper bound on the diameter of some oriented graphs, and in a particular case, related to Caccetta Häggkvist conjecture, we improve a result of Broersma and Li obtained in 2002 (see [H.J. Broersma, X. Li, Some approaches to a conjecture on short cycles in digraphs, Discrete Appl. Math. 120 (2002) 45–53]).     

A Multivariate CLT for Decomposable Random Vectors with Finite Second Moments

Stein's method is used to derive a CLT for dependent random vectors possessing the dependence structure from Barbour et al. J. Combin. Theory Ser. B 47, 125–145, but under the assumption of second moments only. This allows us to derive Lindeberg–Feller type theorems for sums of random vectors with certain dependence structures. We apply the main theorem to the study of three problems: local dependence, random graph degree statistics and finite population statistics. In particular, we consider U-statistics of independent observations as well as of observations drawn without replacement.     

Une combinatoire sous-jacente au théorème des fonctions implicites

Using his theory of combinatorial species, [3.], 1–82 a combinatorial form of the classical multidimensional implicit function theorem. His theorem asserts the existence and (strong) unicity of species satisgying systems of combinatorial equations of a very general type. We present an explicit construction of these species by using a suitable combinatorial version of the Lie Series in the sense of [1. and 2.]. The approach constitutes a generalization of the method of “éclosions” (bloomings) which was used by the author in (J. Combin. Theory Ser. A 39, No. 1 (1985), 52–82), to study multidimensional power series reversion. Remarks concerning the applicability of the method to solve certain combinatorial differential equations are also made at the end of the work.     

Intersection diagrams of distance-biregular graphs

We shall investigate distance-biregular graphs by means of intersection diagrams. First we give an alternate proof of a theorem which was obtained by Mohar and Shawe-Taylor in (J. Combin. Theory Ser. B 37 (1984), 90–100). Next we give some results on distance-biregular graphs of girth g ≡ 0 (mod 4).     

A (c. L*-Geometry for the Sporadic Group J2

We prove the existence of a rank three geometry admitting the Hall–Janko group J2 as flag-transitive automorphism group and Aut(J2) as full automorphism group. This geometry belongs to the diagram (c·L*) and its nontrivial residues are complete graphs of size 10 and dual Hermitian unitals of order 3.     

The edge-diametric theorem in Hamming spaces

The maximum number of edges spanned by a subset of given diameter in a Hamming space with alphabet size at least three is determined. The binary case was solved earlier by Ahlswede and Khachatrian [A diametric theorem for edges, J. Combin. Theory Ser. A 92(1) (2000) 1–16].     

A new short proof of a theorem of Ahlswede and Khachatrian

Ahlswede and Khachatrian [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] proved the following theorem, which answered a question of Frankl and Füredi [P. Frankl, Z. Füredi, Nontrivial intersecting families, J. Combin. Theory Ser. A 41 (1986) 150-153]. Let 2?t+1?k?2t+1 and n?(t+1)(k−t+1). Suppose that F is a family of k-subsets of an n-set, every two of which have at least t common elements. If |_?F∈FF|<t, then , and this is best possible. We give a new, short proof of this result. The proof in [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] requires the entire machinery of the proof of the complete intersection theorem, while our proof uses only ordinary compression and an earlier result of Wilson [R.M. Wilson, The exact bound in the Erd?s-Ko-Rado theorem, Combinatorica 4 (1984) 247-257].     

A geometric consequence of the classification of finite doubly transitive groups

We prove that the finite linear spaces containing a proper linear subspace and admitting an automorphism group which is transitive on the unordered pairs of intersecting lines are the projective and affine spaces of dimension 3, unless all lines have size 2.     

Canonization theorems for finite affine and linear spaces

In this paper we prove a canonical (i.e. unrestricted) version of the Graham—Leeb—Rothschild partition theorem for finite affine and linear spaces [3]. We also mention some other kind of canonization results for finite affine and linear spaces.     

Extending Greene's theorem to directed graphs

The purpose of this note is to prove a theorem on acyclic digraphs, conjectured by Linial (J. Combin. Theory Ser. A, 30 (1981), 331–334), which includes Greene's theorem (J. Combin. Theory Ser. A, 20 (1976), 69–79) on decompositions of a partially ordered set into antichains.     

An algorithmic sign-reversing involution for special rim-hook tableaux

Eğecioğlu and Remmel [Linear Multilinear Algebra 26 (1990) 59–84] gave an interpretation for the entries of the inverse Kostka matrix K⁻¹ in terms of special rim-hook tableaux. They were able to use this interpretation to give a combinatorial proof that KK⁻¹=I but were unable to do the same for the equation K⁻¹K=I. We define an algorithmic sign-reversing involution on rooted special rim-hook tableaux which can be used to prove that the last column of this second product is correct. In addition, following a suggestion of Chow [preprint, math.CO/9712230, 1997] we combine our involution with a result of Gasharov [Discrete Math. 157 (1996) 193–197] to give a combinatorial proof of a special case of the (3+1)-free Conjecture of Stanley and Stembridge [J. Combin. Theory Ser. A 62 (1993) 261–279].