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%I M2092 N0827 #42 Mar 22 2024 18:38:01
%S 0,0,0,2,16,134,1164,10982,112354,1245676,14909340,191916532,
%T 2646100822,38932850396,609137502242,10101955358506,177053463254274,
%U 3270694371428814,63524155236581118,1294248082658393546,27604013493657933856,615135860462018980316
%N Number of permutations of order n with the length of longest run equal to 4.
%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 262. (Terms for n>=13 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="/A001252/b001252.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
%F a(n) ~ c * d^n * n!, where d = 0.9856086571158818186406473023... and c = 1.057499715221728926169821281... - _Vaclav Kotesovec_, Aug 18 2018
%t length = 4;
%t g[u_, o_, t_] := g[u, o, t] = If[u+o == 0, 1, Sum[g[o + j - 1, u - j, 2], {j, 1, u}] + If[t<length, Sum[g[u + j - 1, o - j, t+1], {j, 1, o}], 0]];
%t b[u_, o_, t_] := b[u, o, t] = If[t == length, g[u, o, t], Sum[b[o + j - 1, u - j, 2], {j, 1, u}] + Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]];
%t a[n_] := Sum[b[j - 1, n - j, 1], {j, 1, n}];
%t Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Aug 18 2018, after _Alois P. Heinz_ *)
%Y Cf. A001250, A001251, A001253, A010026, A211318.
%K nonn
%O 1,4
%A _N. J. A. Sloane_
%E Corrected and extended by _Max Alekseyev_ at the suggestion of _Sean A. Irvine_, May 04 2012
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