A000363 Number of permutations of [n] with exactly 2 increasing runs of length at least 2.

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

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

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)

Programs

Magma

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

Mathematica

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

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

Extensions

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