A000402 Number of permutations of [n] in which the longest increasing run has length 3. (Formerly M4239 N1771)

0, 0, 1, 6, 41, 293, 2309, 19975, 189524, 1960041, 21993884, 266361634, 3465832370, 48245601976, 715756932697, 11277786883720, 188135296651083, 3313338641692957, 61444453534759589, 1196988740015236617, 24442368179977776766, 522124104504306695929

Offset

1,4

References

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261, Table 7.4.1. (Values for n>=16 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..452 (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

Example

a(4)=6 because we have (124)3, (134)2, (234)1, 4(123), 3(124) and 2(134), where the parentheses surround increasing runs of length 3.

Mathematica

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, 3];
Array[a, 30] (* Jean-François Alcover, Jul 19 2018, after Alois P. Heinz *)

Crossrefs

Column 3 of A008304. Other columns: A000303, A000434, A000456, A000467.

Cf. A001250, A001251, A001252, A001253, A010026, A211318.

Sequence in context: A196954 A122371 A083067 * A186654 A152107 A143023

Adjacent sequences: A000399 A000400 A000401 * A000403 A000404 A000405

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Better description from Emeric Deutsch, May 08 2004

Terms a(16), a(17) are corrected and further terms added by Max Alekseyev, May 20 2012

Status

approved