A001252 Number of permutations of order n with the length of longest run equal to 4. (Formerly M2092 N0827)

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

Offset

1,4

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

Formula

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

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[t<length, Sum[g[u + j - 1, o - j, t+1], {j, 1, o}], 0]];
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}]];
a[n_] := Sum[b[j - 1, n - j, 1], {j, 1, n}];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Aug 18 2018, after Alois P. Heinz *)

Crossrefs

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

Sequence in context: A037622 A362718 A264333 * A067058 A002302 A348803

Adjacent sequences: A001249 A001250 A001251 * A001253 A001254 A001255

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved