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%I M4735 N2027 #32 Nov 28 2023 13:03:24
%S 0,0,0,0,1,10,99,1024,11304,133669,1695429,23023811,333840443,
%T 5153118154,84426592621,1463941342191,26793750988542,516319125748337,
%U 10451197169218523,221738082618710329,4921234092461339819,114041894068935641488
%N Number of permutations of [n] in which the longest increasing run has length 5.
%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="/A000456/b000456.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], 2012-2013.
%e a(6)=10 because we have (12346)5, (12356)4, (12456)3, (13456)2, (23456)1, 6(12345), 5(12346), 4(12356), 3(12456) and 2(13456), where the parentheses surround increasing runs of length 5.
%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, 5]; Array[a, 25] (* _Jean-François Alcover_, Feb 08 2016, after _Alois P. Heinz_ in A008304 *)
%Y Column 5 of A008304. Other columns: A000303, A000402, A000434, A000467.
%Y Cf. A001250, A001251, A001252, A001253, A010026, A211318.
%K nonn
%O 1,6
%A _N. J. A. Sloane_
%E Better description from _Emeric Deutsch_, May 08 2004
%E Edited and extended by _Max Alekseyev_, May 20 2012
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