A322294 Number of permutations of [n] with exactly floor(n/2) rising or falling successions.

1, 1, 2, 4, 10, 48, 120, 888, 2198, 22120, 54304, 685368, 1674468, 25344480, 61736880, 1087931184, 2644978110, 53138966904, 129019925424, 2909014993080, 7056278570108, 176372774697856, 427516982398576, 11729862804913680, 28417031969575260, 848948339328178128

Offset: 0

Author

Alois P. Heinz, Dec 02 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..696

Crossrefs

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

Cf. A001100.

Programs

Maple

S:= proc(n) option remember; `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)))
    end:
a:= n-> coeff(S(n),t,floor(n/2)):
seq(a(n), n=0..30);

Mathematica

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[n_, k_] := Ceiling[Coefficient[s[n], t, k]];
a[n_] := t[n, Floor[n/2]];
a /@ Range[0, 30] (* Jean-François Alcover, Sep 25 2019, after Alois P. Heinz *)

Formula

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