A000500 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-7 places.

31, 304, 4230, 43880, 547338, 6924960, 94714620, 1375878816, 21273204330, 348919244768, 6056244249682, 110955673493568, 2140465858763844, 43379533256972640, 921616584567907176, 20485188316420940640, 475499882089797554181, 11506280235885243825696

Offset: 7

Author

N. J. A. Sloane

Keywords

• nonn

References

• J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
• 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, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]

Crossrefs

Cf. A000492, A000470, A000440, A000476, A000380, A000388.

Programs

Maple

seq(f(n,7), n=7..30); # code for f(n,k) is given in A000440 - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

Mathematica

sigma[t_, u_] := (1-2*t^2*u^2 - 2*t^2*(1+t)*u^3 + 3*t^4*u^4)/((1-t*u)*(1-(1 + 2*t)*u - t*u^2 + t^3*u^3)); ds[t_, n_] := D[sigma[t, u], {u, n}] /. u -> 0; f[n_, k_] := Coefficient[ Sum[ Coefficient[ds[t, n]/n!, t, j]*(n-j)!*(y-1)^j, {j, 0, n}], y, k]; a[n_] := f[n, 7]; Table[Print[an = a[n]]; an, {n, 7, 24}] (* Jean-François Alcover, Jan 25 2013, after Maple code *)

Formula

a(n) = coefficient of y^7 in sum_0^n sigma_{n, k}(n - k)!(y - 1)^k on y where the sigma_{n, k} have generating function sigma(t, u) = (1 - 2t^2(u^2) - 2t^2(1 + t)u^3 + 3t^4(u^4))(1 - tu)^(-1)(1 - (1 + 2t)u - tu^2 + t^3(u^3))^(-1). - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

Extensions

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001