Let \( n \) be a natural number. We obtain convolution-type formulas for the total number of parts in all partitions of \( n \) of several different kinds.

In this paper, we establish a doubling method to construct inequivalent Hadamard matrices of order \( 2n \), from Hadamard matrices of order \( n \). Our doubling method uses heavily the symmetric group \( S_n \), where \( n \) is the order of a Hadamard matrix. We improve the efficiency of the method by introducing some group-theoretical heuristics. Using the doubling method in conjunction with the standard 4-row profile criterion, we have constructed several millions of new inequivalent Hadamard matrices of orders \(48, 56, 64, 72, 80, 88, 96,\) and several hundreds of inequivalent Hadamard matrices of orders 672 and 856. The Magma code segments, included in this paper, allow one to compute many more inequivalent Hadamard matrices of the above orders and all other orders of the form \( 8t \).

In this paper, we determine analytically the number of balanced, unlabelled, 3-member covers of an unlabelled finite set, which is then used to find the number of non-isomorphic optimal lottery sets of cardinality three. We also determine numerically the number of non-isomorphic optimal playing sets for lotteries in which a single correct number is required to win a prize.

A fire breaks out on a graph \( G \) and then \( f \) firefighters protect \( f \) vertices. At each subsequent interval, the fire spreads to all adjacent unprotected vertices, and firefighters protect \( f \) unburned vertices. Let \( f_G \) be the minimum number of firefighters needed to contain a fire on graph \( G \). If the triangular grid goes unprotected to time \( t = k \) when \( f_G \) firefighters arrive and begin protecting vertices, the fire can be contained by time \( t = 18k + 3 \) with at most \( 172k^2 + 58k + 5 \) vertices burned.

A construction is given for a Restricted Sarvate-Beam Triple System in the case \( v = 8 \). This is the extremal case, since a Restricted SB Triple System cannot exist for \( v > 8 \).

A \( t \)-\((v, k, \lambda) \) covering is a set of blocks of size \( k \) such that every \( t \)-subset of a set of \( v \) points is contained in at least \( \lambda \) blocks. The cardinality of the set of blocks is the size of the covering. The covering number \( C_\lambda(v, k, t) \) is the minimum size of a \( t \)-\((v, k, \lambda) \) covering. In this article, we find upper bounds on the size of \( t \)-\((v, k, 2) \) coverings for \( t = 3, 4 \), \( k = 5, 6 \), and \( v \leq 18 \). Twelve of these bounds are the exact covering numbers.

A tournament \(T = (V, A)\) is \({arc-traceable}\) if for each arc \(xy \in A\), \(xy\) lies on a directed path containing all the vertices of \(V\), i.e., a hamiltonian path. In this paper, we give two extremal results related to arc-traceability in tournaments. First, we show that a non-arc-traceable tournament \(T\) which is \(m\)-arc-strong must have at least \(2^{m+1}+4m-3\) vertices, and we construct an example that shows that this result is best possible. Next, we consider the maximum number of arcs in a strong tournament that are not part of any hamiltonian path. We use the structure of non-arc-traceable tournaments to prove that no strong tournament contains more than \(\frac{n^2-4n+3}{8}\) arcs that are not part of a hamiltonian path, and we give the unique example that shows that this bound is best possible.

We point out that restricted SB triple systems can only exist for \(v \leq 8\). The case \(v = 8\) is especially interesting since it is extremal in that the pair frequencies of the fifteen pairs not involving either \(1\) or \(2\) must be the frequencies \(2, 3, \dots, 16\), in some order.

Let \( a \) and \( b \) be two positive integers. For the graph \( G \) with vertex set \( V(G) \) and edge set \( E(G) \) with \( p = |V(G)| \) and \( q = |E(G)| \), we define two sets \( Q(a) \) and \( P(b) \) as follows:

\[
Q(a) = \begin{cases}
\{\pm a, \pm(a+1), \ldots, \pm(a + (q-2)/2)\} & \text{if } q \text{ is even,} \\
\{0\} \cup \{\pm a, \pm(a+1), \ldots, \pm(a + (q-3)/2)\} & \text{if } q \text{ is odd,}
\end{cases}
\]

\[
P(b) = \begin{cases}
\{\pm b, \pm(b+1), \ldots, \pm(b + (p-2)/2)\} & \text{if } p \text{ is even,} \\
\{0\} \cup \{\pm b, \pm(b+1), \ldots, \pm(b + (p-3)/2)\} & \text{if } p \text{ is odd.}
\end{cases}
\]

For the graph \( G \) with \( p = |V(G)| \) and \( q = |E(G)| \), \( G \) is said to be \( Q(a)P(b) \)-super edge-graceful (in short, \( Q(a)P(b) \)-SEG), if there exists a function pair \( (f, f^+) \) which assigns integer labels to the vertices and edges; that is, \( f^+: V(G) \to P(b) \), and \( f: E(G) \to Q(a) \) such that \( f^+ \) is onto \( P(b) \) and \( f \) is onto \( Q(a) \), and

\[
f^+(u) = \sum\{ f(u,v) : (u, v) \in E(G) \}.
\]

We investigate \( Q(a)P(b) \) super-edge-graceful labelings for three classes of \( (p,p+1) \)-graphs.