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A086855 Number of permutations of length n with exactly 4 rising or falling successions. 3


%S 0,0,0,0,0,2,22,226,2198,22120,236968,2732268,33940644,453148422,

%T 6480322210,98907371822,1605581578202,27631315113916,502618772515748,

%U 9637245372790760,194291040277517688,4109014039030693578,90968013940830446574,2104072961763468757082

%N Number of permutations of length n with exactly 4 rising or falling successions.

%C Permutations of 12...n such that exactly 4 of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).

%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.

%H Alois P. Heinz, <a href="/A086855/b086855.txt">Table of n, a(n) for n = 0..200</a>

%H J. Riordan, <a href="http://projecteuclid.org/euclid.aoms/1177700181">A recurrence for permutations without rising or falling successions</a>, Ann. Math. Statist. 36 (1965), 708-710.

%F Coefficient of t^4 in S[n](t) defined in A002464.

%F a(n) ~ 2/3*exp(-2) * n!. - _Vaclav Kotesovec_, Aug 14 2013

%p S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]

%p [n+1], expand((n+1-t)*S(n-1) -(1-t)*(n-2+3*t)*S(n-2)

%p -(1-t)^2*(n-5+t)*S(n-3) +(1-t)^3*(n-3)*S(n-4)))

%p end:

%p a:= n-> ceil(coeff(S(n), t, 4)):

%p seq (a(n), n=0..25); # _Alois P. Heinz_, Jan 11 2013

%t S[n_] := S[n] = If[n<4, {1, 1, 2*t, 4*t+2*t^2}[[n+1]], Expand[(n+1-t)*S[n-1] - (1-t)*(n-2+3*t)*S[n-2] - (1-t)^2*(n-5+t)*S[n-3] + (1-t)^3*(n-3)*S[n-4]]]; a[n_] := Ceiling[Coefficient[S[n], t, 4]]; Table [a[n], {n, 0, 25}] (* _Jean-Fran├žois Alcover_, Oct 13 2014, after _Alois P. Heinz_ *)

%Y Cf. A002464, A000130, A000349, A001267, A086852, A086853. A diagonal of A001100.

%Y Twice A001268.

%K nonn

%O 0,6

%A _N. J. A. Sloane_, Aug 19 2003

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Last modified December 22 15:37 EST 2014. Contains 252364 sequences.