A000507 Number of permutations of [n] with exactly 3 increasing runs of length at least 2. (Formerly M5323 N2314)

61, 1385, 19028, 206276, 1949762, 16889786, 137963364, 1081702420, 8236142455, 61386982075, 450403628440, 3266265481144, 23480284103492, 167687984079924, 1191656966048088, 8436830209386360, 59563995267159825, 419628657826253805

Offset

6,1

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

Table of n, a(n) for n=6..23.

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 (30, -385, 2776, -12418, 35908, -67818, 82552, -62109, 26190, -4725).

Formula

a(n) = (3*7^n-(6*n-9)*5^n+(6*n^2-18*n+3)*3^n-4*n^3+18*n^2-8*n-15)/192.

G.f.: x^6*(61 - 445*x + 963*x^2 - 675*x^3)/((1 - 5*x)^2*(1 - x)^4*(1 - 3*x)^3*(1 - 7*x)). - Stefano Spezia, Nov 09 2024

Mathematica

t[n_, 0] = 1; t[n_, k_] /; k > n/2 = 0; t[n_, k_] /; k <= n/2 := t[n, k] = (2k+1) t[n-1, k] + (n-2k+1) t[n-1, k-1]; a[n_] := t[n, 3]; Table[a[n], {n, 6, 23}] (* Jean-François Alcover, Feb 09 2016 *)

Prog

(Magma) [(3*7^n-(6*n-9)*5^n+(6*n^2-18*n+3)*3^n-4*n^3+18*n^2-8*n-15)/192: n in [6..30]]; // Vincenzo Librandi, Feb 09 2016

Crossrefs

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

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

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

(End)

Sequence in context: A154428 A262017 A060061 * A350974 A143011 A211213

Adjacent sequences: A000504 A000505 A000506 * A000508 A000509 A000510

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Definition changed for clarity and for consistency with A008971, and formula and additional terms added by Jon E. Schoenfield, Mar 26 2010

Status

approved