Note on $r$-central Lah numbers and $r$-central Lah-Bell numbers

The $r$-Lah numbers generalize the Lah numbers to the $r$-Stirling numbers in the same sense. The Stirling numbers and the central factorial numbers are one of the important tools in enumerative combinatorics. The $r$-Lah number counts the number of partitions of a set with $n+r$ elements into $k+r$ ordered blocks such that $r$ distinguished elements have to be in distinct ordered blocks. In this paper, the $r$-central Lah numbers and the $r$-central Lah-Bell numbers ( $r\in \mathbb{N}$) are introduced parallel to the $r$-extended central factorial numbers of the second kind and $r$-extended central Bell polynomials. In addition, some identities related to these numbers including the generating functions, explicit formulas, binomial convolutions are derived. Moreover, the $r$-central Lah numbers and the $r$-central Lah-Bell numbers are shown to be represented by Riemann integral, respectively.