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A333726 Triangle read by rows: T(n,k) is the number of permutations in S_n for which all cycles are length k or greater.

%I A333726 #26 Feb 17 2022 13:34:35
%S A333726 1,2,1,6,2,2,24,9,6,6,120,44,24,24,24,720,265,160,120,120,120,5040,
%T A333726 1854,1140,720,720,720,720,40320,14833,8988,6300,5040,5040,5040,5040,
%U A333726 362880,133496,80864,58464,40320,40320,40320,40320,40320
%N A333726 Triangle read by rows: T(n,k) is the number of permutations in S_n for which all cycles are length k or greater.
%H A333726 Peter Kagey, <a href="/A333726/b333726.txt">Table of n, a(n) for n = 1..10011</a> (first 141 rows, flattened)
%H A333726 Steven Finch, <a href="http://arxiv.org/abs/2202.07621">Second best, Third worst, Fourth in line</a>, arxiv:2202.07621 [math.CO], 2022.
%F A333726 T(n,k) = 0 when n < k,
%F A333726 T(n,k) = (n-1)! when n = k, and otherwise
%F A333726 T(n,k) = (n-1) * T(n-1, k) + A068424(n-1, k-1) * T(n-k, k).
%e A333726 Triangle begins:
%e A333726 n\k|      1      2     3     4     5     6     7     8     9
%e A333726 ---+--------------------------------------------------------
%e A333726   1|      1
%e A333726   2|      2      1
%e A333726   3|      6      2     2
%e A333726   4|     24      9     6     6
%e A333726   5|    120     44    24    24    24
%e A333726   6|    720    265   160   120   120   120
%e A333726   7|   5040   1854  1140   720   720   720   720
%e A333726   8|  40320  14833  8988  6300  5040  5040  5040  5040
%e A333726   9| 362880 133496 80864 58464 40320 40320 40320 40320 40320
%Y A333726 Columns 1-4: A000142, A000166, A038205, A047865.
%Y A333726 Cf. A068424.
%K A333726 nonn,tabl
%O A333726 1,2
%A A333726 _Peter Kagey_, Apr 23 2020