Let \(G\) be a finite group and \(S \subseteq G \setminus \{0\}\). We call \(S\) an additive basis of \(G\) if every element of \(G\) can be expressed as a sum over a nonempty subset in some order. Let \(cr(G)\) be the smallest integer \(t\) such that every subset of \(G \setminus \{0\}\) of cardinality \(t\) is an additive basis of \(G\). In this paper, we determine \(cr(G)\) for the following cases: (i) \(G\) is a finite nilpotent group; (ii) \(G\) is a group of even order which possesses a subgroup of index \(2\).

For \(n \geq 1\), we let \(a_n\) count the number of compositions of the positive integer \(m\), where the last summand is odd. We find that \(a_n = (\frac{1}{3})(-1)^n + (\frac{2}{3}) 2^{n-1}\). Since \(J_n\), the \(n\)-th Jacobsthal number, is given as \(\frac{1}{3}(-1)^n + \frac{2}{3}2^{n-1}\) for \(n \geq 0\), it follows that \(a_n = J_{n-1}\) for \(n \geq 1\). For this reason, these compositions are often referred to as the Jacobsthal compositions.

In our investigation, we determine results for the \(a_n\) compositions of \(n\), such as: (i) \(a_{n,k}\), the number of times the positive integer \(k\) appears as a summand among these \(a_n\) compositions of \(n\); (ii) the numbers of plus signs, summands, even summands, and odd summands that occur for these compositions of \(n\); (iii) the sum of the even summands and the sum of the odd summands for the \(a_n\) compositions of \(n\); (iv) the numbers of levels, rises, and descents for the \(a_n\) compositions; and (v) the number of runs that occur among these \(a_n\) compositions.

In this paper, we introduce a new sequence called standard Young words, which are defined as quaternary words with interesting restrictions. First, we show that the cardinality of standard Young words of length n is related to Catalan triangle sequence and we establish a bijection from the set of standard Young words to the set of pairs of non-intersection lattice paths. Then we set a one-to-one correspondence between the set of standard Young words and the set of standard Young tableaux of two rows, which results in the correspondence between the statistics of standard Young words and standard Young tableaux, such as sign and descents.

A graph \(G\) is called a fractional \((k, m)\)-deleted graph if after deleting any \(m\) edges of \(G\), the resulting graph admits a fractional \(k\)-factor. In this paper, we prove that for \(k \geq 2\) and \(m \geq 0\), \(G\) is a fractional \((k, m)\)-deleted graph if one of the following conditions holds: 1) \(n \geq 4k + 4m – 3\), \(\delta(G) \geq k + m\), and \(\max\{d_G(u), d_G(v)\} \geq \frac{n}{2}\) for each pair of non-adjacent vertices \(u\) and \(v\) of \(G\); 2) \(\delta(G) \geq k + m\), \(\omega_2(G) \geq n\), \(n \geq 4k + 4m – 5\) if \((k, m) = (3, 0)\), and \(n \geq 8\) if \((k, m) = (3, 0)\). The results are best possible in some sense.

Let \(K\) be a real quadratic field \(\mathbb{Q}(\sqrt{n})\) with an integer \(n = df^2\), where \(d\) is the field discriminant of \(K\) and \(f \geq 1\). Q. Mushtaq found an interesting phenomenon that any totally negative number \(\kappa_0\) with \(\kappa^{\sigma} < 0\) and \(\kappa_0^{\sigma} < 0\) belonging to the discriminant \(n\), attains an ambiguous number \(\kappa_m\) with \(\kappa_m \kappa_m^{\sigma} < 0\) after finitely many actions \(\kappa_0^{A_j}\) with \(0 \leqq j \leqq m\) by modular transformations \(A_j \in \mathrm{SL}_2^+(\mathbb{Z})\). Here \(\sigma\) denotes the embedding of \(K\) distinct from the identity. In this paper, we give a new aspect for the process to reach an ambiguous number from a totally negative or totally positive number, by which the gap of the proof of Q. Mushtaq's Theorem is complemented. Next, as an analogue of Gauss' Genus Theory, we prove that the ring class number \(h_{+}(df^2)\) coincides with the ambiguous class number belonging to the discriminant \(n = df^2\), and its behavior is unbounded when \(f\) with suitable prime factors goes to infinity using the ring class number formula.

For a rational number \(r > 1\), a set \(A\) of positive integers is called an \(r\)-multiple-free set if \(A\) does not contain any solution of the equation \(rx = y\). The extremal problem of estimating the maximum possible size of \(r\)-multiple-free sets contained in \([n] := \{1, 2, \ldots, n\}\) has been studied in combinatorial number theory for theoretical interest and its application to coding theory. Let \(a\) and \(b\) be relatively prime positive integers such that \(a < b\). Wakeham and Wood showed that the maximum size of \((b/a)\)-multiple-free sets contained in \([n]\) is \( \frac{b}{b+1} + O(\log n)\). In this note, we generalize this result as follows. For a real number \(p \in (0, 1)\), let \([n]_p\) be a set of integers obtained by choosing each element \(i \in [n]\) randomly and independently with probability \(p\). We show that the maximum possible size of \((b/a)\)-multiple-free sets contained in \([n]_p\) is \({\frac{b}{b+p}pn} + O(\sqrt{pn} \log n \log \log n)\) with probability that goes to \(1\) as \(n \to \infty\).

A partition of an integer \(n\) is a representation \(n = a_1 + a_2 + \cdots + a_k\), with integer parts \(a_1 \geq a_2 \geq \cdots \geq a_k \geq 1\). The Durfee square is the largest square of points in the graphical representation of a partition. We consider generating functions for the sum of areas of the Durfee squares for various different classes of partitions of \(n\). As a consequence, interesting partition identities are derived. The more general case of Durfee rectangles is also treated, as well as the asymptotic growth of the mean area over all partitions of \(n\).

A graph \(G\) is called a fractional \((k, m)\)-deleted graph if any \(m\) edges are removed from \(G\), then the resulting graph admits a fractional \(k\)-factor. In this paper, we prove that for integers \(k \geq 2\), \(m \geq 0\), \(n \geq 8k + 4m – 7\), and \(\delta(G) \geq k + m\), if
\[|N_G(x) \cup N_G(y)| \geq \frac{n}{2}\]
for each pair of non-adjacent vertices \(x, y\) of \(G\), then \(G\) is a fractional \((k, m)\)-deleted graph. The bounds for neighborhood union condition, order, and the minimum degree of \(G\) are all sharp.

A \(c\)-partite or multipartite tournament is an orientation of a complete \(c\)-partite graph. A digraph \(D\) is cycle complementary if there exist two vertex-disjoint directed cycles \(C\) and \(C’\) such that \(V(D) = V(C) \cup V(C’)\). The global irregularity of a digraph \(D\) is defined by
\[i_g(D) = \max\{\max(d^+(x), d^-(x)) – \min(d^+(y),d^-(y)) \mid x,y \in V(D)\}.\]
If \(i_g(D) = 0\), then \(D\) is regular, and if \(i_g(D) \leq 1\), then \(D\) is almost regular. We prove in this paper that every almost regular \(c\)-partite tournament with \(c \geq 3\) such that all partite sets have the same cardinality \(r \geq 4\) contains two complementary directed cycles of length \(3\) and \(|V(D)| – 3\).

In this paper, we determine the spectrum for \(super-perfect\) OQSs. OQSs are \(G\)-designs in which \(G\) is an octagon quadrangle, i.e., the graph consisting of an \(8\)-cycle \((x_1, x_2, \ldots, x_8)\) with two additional chords: the edges \(\{x_1, x_4\}\) and \(\{x_5, x_6\}\).