For any \(n \geq 2\) we let \(S_n\) be the set of permutations of the set \(\{1,2,\ldots,n\}\). A reduction \(\overline{f}\) on \(S_n\) is a set of functions \(\{f_i : 1 \leq i \leq n\}\) such that \(f_n\) is the identity function on \(\{1,2,\ldots,n-1\}\) and for \(i n_0\), such that \(\phi(n) \leq n\) for all \(n \geq n_0\), and for which \(p \downarrow \phi(n) \downarrow i = p \downarrow i \downarrow n-1\) for all \(n > n_0\), for all \(i \leq n-1\) and for all \(p \in S_n\). And the system is said to be amenable if for every \(n > n_0\) there is an integer \(k < n\) such that, for all \(p \in S_n\), \(p \downarrow k \downarrow n-1 = p \downarrow n-1\). The purpose of this paper is to study faithful reductions and linked reduction systems. We characterize amenable, linked reduction systems by means of two types of liftings by which a reduction on \(S_{n+1}\) can be formed from one on \(S_n\). And we obtain conditions for a reduction system to be faithful. One interesting consequence is that any amenable, linked reduction system which begins with a simple reduction is faithful.

Let \(N\) be a positive integer and let \(\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_l)\) be a partition of \(N\) of length \(l\), i.e., \(\sum_{i=1}^{l}\lambda_i = N\) with parts \(\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_l \geq 1\). Define \(T(\lambda)\) as the partition of \(N\) with parts \(l\), \(\lambda_1 – 1, \lambda_2 – 1, \ldots, \lambda_l – 1\), ignoring any zeros that might occur. Starting with a partition \(\lambda\) of \(N\), we describe Bulgarian Solitaire by repeatedly applying the shift operation \(T\) to obtain the sequence of partitions

\[\lambda, T(\lambda), T^2(\lambda), \ldots\]

We say a partition \(\mathcal{A}\) of \(N\) is \(T\)-cyclic if \(T^i(\mu) = \mu\) for some \(i \geq 1\). Brandt \([2]\) characterized all \(T\)-cyclic partitions for Bulgarian Solitaire. In this paper, we give an inductive proof of Brandt’s result.

Let \(D\) be an acyclic digraph. The competition graph of \(D\) is a graph which has the same vertex set as \(D\) and has an edge between \(x\) and \(y\) if and only if there exists a vertex \(v\) in \(D\) such that \((x, v)\) and \((y, v)\) are arcs of \(D\). For any graph \(G\), \(G\) together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number \(k(G)\) of \(G\) is the smallest number of such isolated vertices.
A hole of a graph is a cycle of length at least \(4\) as an induced subgraph. In \(2005\), Kim \([5]\) conjectured that the competition number of a graph with \(h\) holes is at most \(h + 1\). Though Li and Chang \([8]\) and Kim et al. \([7]\) showed that her conjecture is true when the holes do not overlap much, it still remains open for the case where the holes share edges in an arbitrary way. In order to share an edge, a graph must have at least two holes and so it is natural to start with a graph with exactly two holes. In this paper, the conjecture is proved true for such a graph.

Let \(G\) be a graph with domination number \(\gamma(G)\). A dominating set \(S \subseteq V(G)\) has property \(\mathcal{UK}\) if all components of the subgraph it induces in \(G\) are complete. The union of complete graphs domination number of a graph \(G\), denoted \(\gamma_{uk}(G)\), is the minimum possible size of a dominating set of \(G\), which has property \(\mathcal{UK}\). Results on changing and unchanging of \(\gamma_{uk}(G)\) after vertex removal are presented. Also forbidden subgraph conditions sufficient to imply \(\gamma(G) = \gamma_{uk}(G)\) are given.

In this paper, we use the finite Heine \({}_{2}\Phi_1\) transformations given in \([4]\) and some elementary simplifications to obtain several Rogers-Ramanujan type identities.

Using lines in a two-dimensional vector space \({GF}(q^2)\) over \({GF}(q)\), we construct some classes of external difference families over \({GF}(q^2)\).

A digraph \(D\) is a local out-tournament if the outset of every vertex is a tournament. Here, we use local out-tournaments, whose strong components are upset tournaments, to explore the corresponding ranks of the adjacency matrices. Of specific interest is the out-tournament whose adjacency matrix has boolean, nonnegative integer, term, and real rank all equal to the number of vertices, \(n\). Corresponding results for biclique covers and partitions of the digraph are provided.