A002630 Number of permutations of length n with two 3-sequences.

0, 0, 0, 1, 2, 12, 71, 481, 3708, 32028, 306723, 3228804, 37080394, 461569226, 6192527700, 89102492915, 1369014167140, 22373840093040, 387602212164321, 7095737193164187, 136885937242792752, 2775675888994318366, 59023506305591628101, 1313445236142071926488

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• D. M. Jackson, J. W. Reilly, Permutations with a prescribed number of $p$-runs. Ars Combinatoria 1 (1976), no. 1, 297-305.
• 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

• J. Riordan, Permutations without 3-sequences, Bull. Amer. Math. Soc., 51 (1945), 745-748.

Crossrefs

Cf. A047921.

Programs

Mathematica

nmax = 22;
CoefficientList[Sum[((m + 2)*(m + 1)*(m + 2)!/2*(x^6*(1 - x)^4/(1 - x^3)^4) + (m + 1)*(m + 1)!*(x^4*(1 + x)*(1 - x)^3)/(1 - x^3)^3)*((x - x^3)/(1 - x^3))^m, {m, 0, nmax}]/x + O[x]^nmax, x] (* Jean-François Alcover, May 06 2024, after Tani Akinari *)

PARI

concat([0,0,0],Vec(sum(m=0,100,((m+2)*(m+1)*(m+2)!/2*(x^6*(1-x)^4/(1-x^3)^4)+(m+1)*(m+1)!*(x^4*(1+x)*(1-x)^3)/(1-x^3)^3)*((x-x^3)/(1-x^3))^m)+O(x^100))) \\ Tani Akinari, Nov 08 2014

Extensions

More terms from Max Alekseyev, Feb 20 2010