A000303 Number of permutations of [n] in which the longest increasing run has length 2.

0, 1, 4, 16, 69, 348, 2016, 13357, 99376, 822040, 7477161, 74207208, 797771520, 9236662345, 114579019468, 1516103040832, 21314681315997, 317288088082404, 4985505271920096, 82459612672301845, 1432064398910663704, 26054771465540507272

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			a(3)=4 because we have (13)2, 2(13), (23)1, 3(12), where the parentheses surround increasing runs of length 2.
		

References

• F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261, Table 7.4.1.
• 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..464 (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

Column 2 of A008304. Other columns: A000402, A000434, A000456, A000467, A230055.

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

Equals 1 less than A049774. - Greg Dresden, Feb 22 2020

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, 2];
Array[a, 30] (* Jean-François Alcover, Jul 19 2018, after Alois P. Heinz *)

Extensions

Better description from Emeric Deutsch, May 08 2004

Edited and extended by Max Alekseyev, May 20 2012