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{\bf Toufik Mansour, Matthias Schork and Mark Shattuck}
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{\bf On a New Family of Generalized Stirling and Bell Numbers}
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A new family of generalized Stirling and Bell numbers is introduced by
considering powers $(VU)^n$ of the noncommuting variables $U,V$
satisfying $UV=VU+hV^s$. The case $s=0$ (and $h=1$) corresponds to the
conventional Stirling numbers of second kind and Bell numbers. For
these generalized Stirling numbers, the recursion relation is given
and explicit expressions are derived. Furthermore, they are shown to
be connection coefficients and a combinatorial interpretation in terms
of statistics is given. It is also shown that these Stirling numbers
can be interpreted as $s$-rook numbers introduced by Goldman and
Haglund. For the associated generalized Bell numbers, the recursion
relation as well as a closed form for the exponential generating
function is derived. Furthermore, an analogue of Dobinski's formula is
given for these Bell numbers.
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