Abstract
We provide asymptotic expansions for the Stirling numbers of the first kind and,
more generally, the Ewens (or Karamata-Stirling) distribution. Based on these expansions, we obtain some new results on the asymptotic properties of the mode and the maximum of the Stirling numbers and the Ewens distribution. For arbitrary > 0 and
for all sufficiently large n ∈ N, the unique maximum of the Ewens probability mass function
Ln(k) = k ( + 1) · · · ( + n − 1)
n
k
, k = 1, . . . , n,
is attained at k = ⌊an⌋ or ⌈an⌉, where an = log n−􀀀′()/􀀀()−1/2. We prove that the mode is the nearest integer to an for a set of n’s of asymptotic density 1, yet this formula is not true for infinitely many n’s.
Original language | English |
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Article number | 16.8.8 |
Number of pages | 17 |
Journal | Journal of Integer Sequences |
Volume | 19 |
Issue number | 8 |
Publication status | Published - 21 Nov 2016 |
Keywords
- Stirling number of the first kind
- mode of a distribution
- asymptotic expansion
- Ewens distribution