Geodesic
On a new family of generalized Stirling and Bell numbers
The electronic journal of combinatorics, Tome 18 (2011) no. 1
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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.
DOI : 10.37236/564
Classification : 05A15, 05A18, 05A19, 11B37, 11B73, 11B75 
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     author = {Toufik Mansour and Matthias Schork and Mark Shattuck},
     title = {On a new family of generalized {Stirling} and {Bell} numbers},
     journal = {The electronic journal of combinatorics},
     year = {2011},
     volume = {18},
     number = {1},
     doi = {10.37236/564},
     zbl = {1217.05030},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/564/}
}
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Toufik Mansour; Matthias Schork; Mark Shattuck. On a new family of generalized Stirling and Bell numbers. The electronic journal of combinatorics, Tome 18 (2011) no. 1. doi: 10.37236/564

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