A001252 Number of permutations of order n with the length of longest run equal to 4.

0, 0, 0, 2, 16, 134, 1164, 10982, 112354, 1245676, 14909340, 191916532, 2646100822, 38932850396, 609137502242, 10101955358506, 177053463254274, 3270694371428814, 63524155236581118, 1294248082658393546, 27604013493657933856, 615135860462018980316

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• 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.)
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..450 (first 100 terms from Max Alekseyev)
• Max A. Alekseyev, On the number of permutations with bounded runs length, arXiv preprint arXiv:1205.4581 [math.CO], 2012-2013. - From _N. J. A. Sloane_, Oct 23 2012

Crossrefs

Cf. A001250, A001251, A001253, A010026, A211318.

Programs

Mathematica

length = 4;
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[tJean-François Alcover, Aug 18 2018, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n * n!, where d = 0.9856086571158818186406473023... and c = 1.057499715221728926169821281... - Vaclav Kotesovec, Aug 18 2018

Extensions

Corrected and extended by Max Alekseyev at the suggestion of Sean A. Irvine, May 04 2012