A000363 Number of permutations of [n] with exactly 2 increasing runs of length at least 2. (Formerly M4018 N1666)

5, 61, 479, 3111, 18270, 101166, 540242, 2819266, 14494859, 73802835, 373398489, 1881341265, 9453340172, 47417364268, 237571096820, 1189405165908, 5951965440609, 29775517732665, 148927275340835, 744793282001995

Offset

4,1

References

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.

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

Vincenzo Librandi, Table of n, a(n) for n = 4..1000

Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See p. 7.

Index entries for linear recurrences with constant coefficients, signature (14,-75,196,-263,174,-45).

Formula

From Vaclav Kotesovec, Nov 19 2012: (Start)

a(n) = (5^n-(2*n-1)*3^n+2*n^2-2*n-2)/16.

G.f.: -x^4*(9*x-5)/((x-1)^3*(3*x-1)^2*(5*x-1)). (End)

E.g.f.: exp(x)*(exp(4*x) + exp(2*x)*(1 - 6*x) - 2*(1 - x^2))/16. - Stefano Spezia, Nov 09 2024

Mathematica

Table[(5^n-(2*n-1)*3^n+2*n^2-2*n-2)/16, {n, 4, 20}] (* Vaclav Kotesovec, Nov 19 2012 *)

Prog

(Magma) [(5^n-(2*n-1)*3^n+2*n^2-2*n-2)/16: n in [4..30]]; // Vincenzo Librandi, May 03 2013

Crossrefs

Contribution from Johannes W. Meijer, May 24 2009: (Start)

The a(n) sequence equals the third left hand column of A008971.

The a(2*n) sequence equals the third left hand column of A160486.

(End)

Sequence in context: A060060 A183379 A345649 * A200956 A203358 A099666

Adjacent sequences: A000360 A000361 A000362 * A000364 A000365 A000366

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms and better definition from Jon E. Schoenfield, Mar 25 2010

Status

approved