A000456 Number of permutations of [n] in which the longest increasing run has length 5.

0, 0, 0, 0, 1, 10, 99, 1024, 11304, 133669, 1695429, 23023811, 333840443, 5153118154, 84426592621, 1463941342191, 26793750988542, 516319125748337, 10451197169218523, 221738082618710329, 4921234092461339819, 114041894068935641488

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			a(6)=10 because we have (12346)5, (12356)4, (12456)3, (13456)2, (23456)1, 6(12345), 5(12346), 4(12356), 3(12456) and 2(13456), where the parentheses surround increasing runs of length 5.
		

References

• F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261.
• 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 run lengths, arXiv preprint arXiv:1205.4581 [math.CO], 2012-2013.

Crossrefs

Column 5 of A008304. Other columns: A000303, A000402, A000434, A000467.

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

Programs

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, 5]; Array[a, 25] (* Jean-François Alcover, Feb 08 2016, after Alois P. Heinz in A008304 *)

Extensions

Better description from Emeric Deutsch, May 08 2004

Edited and extended by Max Alekseyev, May 20 2012