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

In this paper, we define the q-analogue of the so-called symmetric infinite matrix algorithm. We find an explicit formula for entries in the associated matrix and also for the generating function of the k-th row of this matrix for each fixed k. This helps us to derive analytic and number theoretic identities with respect to the q-harmonic numbers and q-hyperharmonic numbers of Mansour and Shattuck.

Bargraphs are lattice paths in \(\mathbb{N}_0^2\) with three allowed types of steps: up \((0,1)\), down \((0,-1)\), and horizontal \((1,0)\). They start at the origin with an up step and terminate immediately upon return to the \(x\)-axis. A wall of size \(r\) is a maximal sequence of \(r\) adjacent up steps. In this paper, we develop the generating function for the total number of walls of fixed size \(r \geq 1\). We then derive asymptotic estimates for the mean number of such walls.

We survey four instances of the Fourier analytic ‘transference principle’or ‘dense model lemma’, which allows one to approximate an unbounded function on the integers by a bounded function with similar Fourier transform. Such a result forms a component of a general method pioneered by Green to count solutions to a single linear equation in a sparse subset of integers.

Let \( E \subseteq \mathbb{F}_q^2 \) be a set in the 2-dimensional vector space over a finite field with \( q \) elements, which satisfies \(|E| > q\). There exist \( x, y \in E \) such that \(|E \cdot (y – x)| > q/2\). In particular, \( (E + E) \cdot (E – E) = \mathbb{F}_q.\)

Let \((x(n))_{n \geq 1}\) be an \(s\)-dimensional Niederreiter-Xing sequence in base \(b\). Let \(D((x(n))_{n=1}^N)\) be the discrepancy of the sequence \((x(n))_{n=1}^N\). It is known that \(ND((x(n))_{n=1}^N) = O(\ln^s N)\) as \(N \to \infty\). In this paper, we prove that this estimate is exact. Namely, there exists a constant \(K > 0\), such that
\[
\inf_{w \in [0,1]^s} \sup_{1 \leq N \leq b^m} ND((x(n) \oplus w)_{n=1}^N) \geq K \ln^s \quad \text{ for } m = 1, 2, \ldots.
\]

We also get similar results for other explicit constructions of \((t,s)\)-sequences.