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%I M4556 N1938 #30 Feb 01 2022 00:56:05
%S 0,0,0,1,8,67,602,5811,60875,690729,8457285,111323149,1569068565,
%T 23592426102,377105857043,6387313185576,114303481217657,
%U 2155348564847332,42719058006864690,887953677898186108,19316200230609433690,438920223893512987430
%N Number of permutations of [n] in which the longest increasing run has length 4.
%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261. (Values for n>=16 are incorrect.)
%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="/A000434/b000434.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 runs length</a>, arXiv preprint arXiv:1205.4581 [math.CO], 2012-2013. - From _N. J. A. Sloane_, Oct 23 2012
%e a(5)=8 because we have (1235)4, (1245)3, (1345)2, (2345)1, 5(1234), 4(1235), 3(1245) and 2(1345), where the parentheses surround increasing runs of length 4.
%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 T[n_, k_] := b[0, n, 0, k] - b[0, n, 0, k + 1];
%t a[n_] := T[n, 4];
%t Array[a, 30] (* _Jean-François Alcover_, Jul 19 2018, after _Alois P. Heinz_ *)
%Y Column 4 of A008304. Other columns: A000303, A000402, A000456, A000467.
%Y Cf. A001250, A001251, A001252, A001253, A010026, A211318.
%K nonn
%O 1,5
%A _N. J. A. Sloane_
%E Better description from _Emeric Deutsch_, May 08 2004
%E Terms a(16)-a(18) corrected and further terms added by _Max Alekseyev_, May 20 2012
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