Abstract
ABSTRACT Based on the combinatorial interpretation of the ordered Bell numbers, which count all the ordered partitions of the set [n] = {1, 2, … n}, this paper introduces the deranged partition as a free-fixed-block permutation of its blocks, then defines the deranged Bell numbers that count the total number of the deranged partitions of [n]. At first, we study the classical properties of these numbers (generating function, explicit formula, convolutions, etc.), we then present an asymptotic behavior of the deranged Bell numbers. Finally, we briefly review some results regarding the r-extension of these numbers by considering that the r first elements must be in distinct blocks.
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