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Number of permutations of [n] with exactly floor(n/2) rising or falling successions.
4

%I #12 Sep 25 2019 03:30:23

%S 1,1,2,4,10,48,120,888,2198,22120,54304,685368,1674468,25344480,

%T 61736880,1087931184,2644978110,53138966904,129019925424,

%U 2909014993080,7056278570108,176372774697856,427516982398576,11729862804913680,28417031969575260,848948339328178128

%N Number of permutations of [n] with exactly floor(n/2) rising or falling successions.

%H Alois P. Heinz, <a href="/A322294/b322294.txt">Table of n, a(n) for n = 0..696</a>

%F a(n) = A001100(n,floor(n/2)).

%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-> coeff(S(n),t,floor(n/2)):

%p seq(a(n), n=0..30);

%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]]];

%t t[n_, k_] := Ceiling[Coefficient[s[n], t, k]];

%t a[n_] := t[n, Floor[n/2]];

%t a /@ Range[0, 30] (* _Jean-François Alcover_, Sep 25 2019, after _Alois P. Heinz_ *)

%Y Bisections give A322295 (even part), A322295 (odd part).

%Y Cf. A001100.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Dec 02 2018