In this paper, we first give a new \( q \)-analogue of the Lah numbers. Then we show the irreducible factors of the \( q \)-Lah numbers over \( \mathbb{Z} \).

Let \( A \) and \( B \) be additive sets of \( \mathbb{Z}_{2k} \), where \( A \) has cardinality \( k \) and \( B = v \cdot C A \) with \( v \in \mathbb{Z}_{2k}^\times \). In this note, some bounds for the cardinality of \( A + B \) are obtained using four different approaches. We also prove that in a special case, the bound is not sharp and we can recover the whole group as a sumset.

In this paper, we analyze the asymptotic number \( I(m,n) \) of involutions of large size \( n \) with \( m \) singletons. We consider a central region and a non-central region. In the range \( m = n – n^\alpha \), \( 0 < \alpha < 1 \), we analyze the dependence of \( I(m,n) \) on \( \alpha \). This paper fits within the framework of Analytic Combinatorics.

An inverse-conjugate composition of a positive integer \(m\) is an ordered partition of \(m\) whose conjugate coincides with its reversal. In this paper, we consider inverse-conjugate compositions in which the part sizes do not exceed a given integer \(k\). It is proved that the number of such inverse-conjugate compositions of \(2n – 1\) is equal to \(2F_n^{(k-1)}\), where \(F_n^{(k)}\) is a Fibonacci \(k\)-step number. We also give several connections with other types of compositions, and obtain some analogues of classical combinatorial identities.

S. Ekhad and D. Zeilberger recently proved that the multivariate generating function for the number of simple singular vector tuples of a generic \(m_1 \times · · · \times m_d\) tensor has an elegant rational form involving elementary symmetric functions, and provided a partial conjecture for the asymptotic behavior of the cubical case \(m_1 = · · · = m_d\). We prove this conjecture and further identify completely the dominant asymptotic term, including the multiplicative constant. Finally, we use the method of differential approximants to conjecture that the subdominant connective constant effect observed by Ekhad and Zeilberger for a particular case in fact occurs more generally