Let \(n,s\) be positive integers, and let \(r = 1 + \frac{1}{s}\). In this note we prove that if the sequence \(\{a_n(r)\}_{n=1}^{\infty}\) satisfies \(ra_n(r)= \sum_{k=1}^{n}\binom{n}{k}a_k(r), n> 1\), then \(a_n(r) = na_1(r)\left[(n -1)! / {(s+1)}(log r)^n+{{1/r(s+1)}} \right]\). Moreover, we obtain a related combinatorial identity.

A Mendelsohn triple system, \(MTS(v) = (X, \mathcal{B})\), is called self-converse if it and its converse \((X, \mathcal{B}^{-1})\) are isomorphic, where \(\mathcal{B}^{-1 } = \{\langle z,y,x\rangle; \langle x,y,z\rangle \in \mathcal{B}\}\). In this paper, the existence spectrum of self-converse \(MTS(v)\) is given, which is \(v \equiv 0\) or \(1 \pmod{3}\) and \(v \neq 6\).

In this paper, we discuss the automorphism groups of circulant digraphs. Our main purpose is to determine the full automorphism groups of circulant digraphs of degree \(3\).

The spectrum for the decomposition of \(\lambda K_v\) into \(3\)-perfect \(9\)-cycles is found for all \(\lambda > 1\). (The case \(\lambda = 1\) was dealt with in an earlier paper by the authors and Lindner.) The necessary conditions for the existence of a suitable decomposition turn out to be sufficient.

A directed triple system of order \(v\), denoted by \(DTS(v)\), is called \((f,k)\)-rotational if it has an automorphism consisting of \(f\) fixed points and \(k\) cycles each of length \((v-f)/k\). In this paper, we obtain a necessary and sufficient condition for the existence of \((f,k)\)-rotational \(DTS(v)\) for any arbitrary positive integer \(k\).

Let \( {R} = (r_1, r_2, \ldots, r_m)\) and \( {S} = (s_1, s_2, \ldots, s_n)\) be nonnegative integral vectors. Denote by \( {A}( {R}, {S})\) the class of \((0,1)\) matrices with row sum vector \( {R}\) and column sum vector \( {S}\). We study a generalization of invariant positions called locally invariant positions of a class \( {A}( {R}, {S})\). For a normalized class, locally invariant positions have in common with invariant positions the property that they lie above and to the left of some simple rook path through the set of positions.

This paper examines the numbers of lattice paths of length \(n\) from the origin to integer points along the line \((a,b,c,d) + t(1,-1,1,-1)\). These numbers form a sequence which this paper shows is log concave, and for sufficiently large values of \(n\), the location of the maximum of this sequence is shown. This paper also shows unimodality of such sequences for other lines provided that \(n\) is sufficiently large.

A cover of a finite set \(N\) is a collection of subsets of \(N\) whose union is \(N\). We determine the number of such covers whose blocks all have distinct sizes. The cases of unordered and ordered blocks are each considered.

Let \(n(k)\) be the smallest number of vertices of a bipartite graph not being \(k\)-choosable. We show that \(n(3) = 14\) and moreover that \(n(k) \leq k. n(k-2)+2^k\). In particular, it follows that \(n(4) \leq 40\) and \(n(6) \leq 304\).

Eight new codes are presented which improve the bounds on maximum minimum distance for binary linear codes. They are rate \(\frac{m-r}{pm},r\geq 1\) , \(r\)-degenerate quasi-cyclic codes.

A method for synthesizing combinatorial structures which are members of an extended class of resolvable incomplete lattice designs is presented. Square and rectangular lattices both are realizable, yet designs in the extended class are not limited in number of treatments by the classically severe restriction \(v = s^2\) or \(v = s(s-1)\). Rather, the current restriction is the condition that there exist a finite closable set of \(k\)-permutations on the objects of some group or finite field, which is then used as the generating array for a \(L(0,1)\) lattice design. A connection to Hadamard matrices \(H(p,p)\) is considered.

Near-perfect protection is a useful extension of perfect protection which is a necessary condition for authentication systems that satisfy Pei-Rosenbaum’s bound. Near-perfect protection implies perfect protection for key strategies, defined in the paper, in which the enemy tries to guess the correct key. We prove a bound on the probability of deception for key strategies, characterize codes that satisfy the bound with equality and conclude the paper with a comparison of this bound and Pei-Rosenbaum’s bound.

This note gives what is believed to be the first published example of a symmetric \(11 \times 11\) Latin square which, although not cyclic, has the property that the permutation between any two rows is an \(11\)-cycle. The square has the further property that two subsets of its rows constitute \(5 \times 11\) Youden squares. The note shows how this \(11 \times 11\) Latin square can be obtained by a general construction for \(n \times n\) Latin squares where \(n\) is prime with \(n \geq 11\). The permutation between any two rows of any Latin square obtained by the general construction is an \(n\)-cycle; two subsets of \((n-1)/2\) rows from the Latin square constitute Youden squares if \(n \equiv 3 \pmod{8}\).

The twenty-five year old \(\lambda\)-design conjecture remains unsettled. Attempts to characterize these irregular, tight, \(2\)-designs have produced a great number of parametric and dual structure characterizations of the so-called Type-I Designs. We establish some new structural characterizations and establish the conjecture in the smallest unsettled case (\(\lambda = 14\)) of the \(2p\) family.

In this paper we consider a random walk in a plane in which a particle at any stage moves one unit in any one of the four directions, namely, north, south, east, and west with equal probability and derive the joint and marginal distributions of certain characteristics of this random walk by using combinatorial methods.

A subset \(S\) of an ordered set \(P\) is called a cutset if each maximal chain of \(P\) has nonempty intersection with \(S\); if, in addition, \(S\) is also an antichain, it is an antichain cutset. We consider new characterizations and generalizations of these and related concepts. The main generalization is to make our definitions in graph theoretic terms. For instance, a cutset is a subset \(S\) of the vertex set \(V\) of graph \(G = (V, E)\) which meets each extremal path of \(G\). Our principal results include (1)a characterization, by means of a closure property, of those antichains which are cutsets;(2) a characterization, by means of “forbidden paths” in the graph, of those graphs which can be expressed as the union of antichain cutsets;(3) a simpler proof of an existing result about \(N\)-free orders; and (4) efficient algorithms for many related problems, such as constructing antichain cutsets containing or excluding specified elements or forming a chain.

We include a brief discussion of the use of antichain cutsets in a parsing problem for \(LR(k)\) languages.

The \(n\)-star graph \(S_n\) is a simple graph whose vertex set is the set of all \(n!\) permutations of \(\{1,2,\ldots,n\}\) and two vertices \(\alpha\) and \(\beta\) are adjacent if and only if \(\alpha(1) \neq \beta(1)\) and \(\alpha(i) \neq \beta(i)\) for exactly one \(i\), \(i \neq 1\). In the paper, we determine the values of the domination number \(\gamma\), the independent domination number \(\gamma_i\), the perfect domination number \(\gamma_p\), and we obtain bounds for the total domination number \(\gamma_t\) and the connected domination number \(\gamma_c\) for \(S_n\).

Holey factorizations of \(K_{v_1,v_2,\ldots,v_n}\) are a basic building block in the construction of Room frames. In this paper we give some necessary conditions for the existence of holey factorizations and give a complete enumeration for nonisomorphic sets of orthogonal holey factorizations of several special types.

It is shown that the maximal number of pairwise edge disjoint forests, \(F\), of order six in the complete graph \(K_n\), and the minimum number of forests of order six, whose union is \(K_n\) are \(\lfloor\frac{n(n-1)}{2e(F)}\rfloor\) and \(\lceil\frac{n(n-1)}{2e(F)}\rceil\), \(n\geq 6\), respectively and \(e(F)\) is the number of edges of \(F\). (\(\lfloor x\rfloor\) denotes the largest integer not exceeding \(x\) and \(\lceil x\rceil\) the least integer not less than \(x\)). Some generalizations to multiple copies of these forests and of paths are also given.

We study four \(q\)-series. Each of which is interpreted combinatorially in three different ways. This results in four new classes of infinite \(3\)-way partition identities. In some particular cases we get even \(4\)-way partition identities. Our every \(3\)-way identity gives us three Roderick-Ramanujan type identities and \(4\)-way identity gives six. Several partition identities due to Gordon \((1965)\), Hirschhorn \((1979)\), Subbarao \((1985)\), Blecksmith et al. \((1985)\), Agarwal \((1988)\) and Subbarao and Agarwal (1988) are obtained as particular cases of our general theorems.

Motivated by the spectral radius of a graph, we introduce the notion of numerical radius for multigraphs and directed multigraphs, and it is proved that, unlike the spectral radius, the numerical radius is invariant under changes in the orientation of a directed multigraph. An analogue of the Perron-Frobenius theorem is given for the numerical radius of a matrix with nonnegative entries.

We consider the polytope \(\mathcal{P}(s)\) of generalized tournament matrices with score vector \(s\). For the case that \(s\) has integer entries, we find the extreme points of \(\mathcal{P}(s)\) and discuss the graph-theoretic structure of its \(1\)-skeleton.