Some formulas for the restricted r-Lah numbers

The r-Lah numbers, which we denote here by `(r)(n, k), enumerate partitions of an (n+r)-element set into k+r contents-ordered blocks in which the smallest r elements belong to distinct blocks. In this paper, we consider a restricted version `(r) m (n, k) of the r-Lah numbers in which block cardinalities are at most m. We establish several combinatorial identities for `(r) m (n, k) and obtain as limiting cases for large m analogous formulas for `(r)(n, k). Some of these formulas correspond to previously established results for `(r)(n, k), while others are apparently new also in the r-Lah case. Some generating function formulas are derived as a consequence and we conclude by considering a polynomial generalization of `(r) m (n, k) which arises as a joint distribution for two statistics defined on restricted r-Lah distributions. Keywords: restricted Lah numbers, polynomial generalization, r-Lah numbers, combinatorial identities MSC: 11B73, 05A19, 05A1

Elliptic rook and file numbers

Utilizing elliptic weights, we construct an elliptic analogue of rook numbers for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's q-rook numbers by two additional independent parameters a and b, and a nome p. These are shown to satisfy an elliptic extension of a factorization theorem which in the classical case was established by Goldman, Joichi and White and later was extended to the q-case by Garsia and Remmel. We obtain similar results for our elliptic analogues of Garsia and Remmel's q-file numbers for skyline boards. We also provide an elliptic extension of the j-attacking model introduced by Remmel and Wachs. Various applications of our results include elliptic analogues of (generalized) Stirling numbers of the first and second kind, Lah numbers, Abel numbers, and r-restricted versions thereof.Comment: 45 pages; 3rd version shortened (elliptic rook theory for matchings has been taken out to keep the length of this paper reasonable

A generalization of the "probléme des rencontres"

In this paper, we study a generalization of the classical \emph{probl\'eme des rencontres} (\emph{problem of coincidences}), consisting in the enumeration of all permutations \pi \in \SS_n with $k$ fixed points, and, in particular, in the enumeration of all permutations \pi \in \SS_n with no fixed points (derangements). Specifically, we study this problem for the permutations of the $n+m$ symbols $1$, $2$, \ldots, $n$, $v_1$, $v_2$, \ldots, $v_m$, where $v_i \not\in\{1,2,\ldots,n\}$ for every $i=1,2,\ldots,m$. In this way, we obtain a generalization of the derangement numbers, the rencontres numbers and the rencontres polynomials. For these numbers and polynomials, we obtain the exponential generating series, some recurrences and representations, and several combinatorial identities. Moreover, we obtain the expectation and the variance of the number of fixed points in a random permutation of the considered kind. Finally, we obtain some asymptotic formulas for the generalized rencontres numbers and the generalized derangement numbers