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%I M4868 N2083 #36 Feb 01 2022 00:58:22
%S 0,0,0,0,0,1,12,137,1602,19710,257400,3574957,52785901,827242933,
%T 13730434111,240806565782,4452251786946,86585391630673,
%U 1767406549387381,37790452850585180,844817788372455779,19711244788916894489,479203883157602851294
%N Number of permutations of [n] in which the longest increasing run has length 6.
%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A000467/b000467.txt">Table of n, a(n) for n = 1..450</a> (first 100 terms from Max Alekseyev)
%H Max A. Alekseyev, <a href="http://arxiv.org/abs/1205.4581">On the number of permutations with bounded run lengths</a>, arXiv preprint arXiv:1205.4581 [math.CO], 2012.
%t b[u_, o_, t_, k_] := b[u, o, t, k] = If[t == k, (u + o)!, If[Max[t, u] + o < k, 0, Sum[b[u + j - 1, o - j, t + 1, k], {j, 1, o}] + Sum[b[u - j, o + j - 1, 1, k], {j, 1, u}]]]; T[n_, k_] := b[0, n, 0, k] - b[0, n, 0, k + 1]; a[n_] := T[n, 6]; Array[a, 23] (* _Jean-François Alcover_, Feb 08 2016, after _Alois P. Heinz_ in A008304 *)
%Y Column 6 of A008304. Other columns: A000303, A000402, A000434, A000456.
%Y Cf. A001250, A001251, A001252, A001253, A010026, A211318.
%K nonn
%O 1,7
%A _N. J. A. Sloane_
%E Edited and extended by _Max Alekseyev_, May 20 2012
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