A187248
Number of permutations of [n] having only cycles with at least 3 alternating runs (it is assumed that the smallest element of the cycle is in the first position).
Original entry on oeis.org
1, 0, 0, 0, 2, 16, 104, 688, 5116, 44224, 438560, 4851136, 58603544, 764606016, 10715043104, 160692920256, 2570016145680, 43678554864128, 786135111482112, 14936420654180864, 298733557232591136, 6273502048592506112, 138018105454095739008, 3174423293668325353216
Offset: 0
a(4)=2 because we have (1423) and (1324).
-
g := exp((1-2*z-exp(2*z))*1/4)/(1-z): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 23);
# second Maple program:
a:= proc(n) option remember;
`if`(n=0, 1, add(a(n-j)*binomial(n-1, j-1)*
`if`(j=1, 0, (j-1)!-2^(j-2)), j=1..n))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Apr 15 2017
-
a[n_] := a[n] = If[n == 0, 1, Sum[a[n-j]*Binomial[n-1, j-1]* If[j == 1, 0, (j-1)! - 2^(j-2)], {j, 1, n}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 16 2018, after Alois P. Heinz *)
A187249
Number of cycles with at most 2 alternating runs in all permutations of [n] (it is assumed that the smallest element of the cycle is in the first position).
Original entry on oeis.org
0, 1, 3, 11, 48, 248, 1504, 10560, 84544, 761024, 7610496, 83715968, 1004592640, 13059706368, 182835893248, 2742538406912, 43880614526976, 745970446991360, 13427468045910016, 255121892872421376, 5102437857448689664, 107151195006423007232, 2357326290141307207680
Offset: 0
a(3)=11 because in (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), and (132) all cycles have at most 2 alternating runs.
-
g := (1/4*(exp(2*z)-1+2*z))/(1-z): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 22);
# second Maple program:
b:= proc(n) option remember; expand(
`if`(n=0, 1, add(b(n-j)*binomial(n-1, j-1)*
`if`(j=1, x, (j-1)!+2^(j-2)*(x-1)), j=1..n)))
end:
a:= n-> (p-> add(coeff(p, x, i)*i, i=0..n))(b(n)):
seq(a(n), n=0..30); # Alois P. Heinz, Apr 15 2017
-
CoefficientList[Series[(E^(2*x)-1+2*x)/(4*(1-x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Mar 15 2014 *)
A187250
Triangle read by rows: T(n,k) is the number of permutations of [n] having k cycles with at least 3 alternating runs (it is assumed that the smallest element of a cycle is in the first position), 0<=k<=floor(n/4).
Original entry on oeis.org
1, 1, 2, 6, 22, 2, 94, 26, 460, 260, 2532, 2508, 15420, 24760, 140, 102620, 254968, 5292, 739512, 2760432, 128856, 5729192, 31547344, 2640264, 47429896, 381339368, 50186136, 46200, 417429800, 4879612808, 926494712, 3483480, 3888426512, 66107044176, 17025751600, 157068912
Offset: 0
T(4,1)=2 because we have (1324) and (1423).
Triangle starts:
1;
1;
2;
6;
22,2;
94,26;
460,260;
-
G := exp((1/4*(1-t))*(2*z-1+exp(2*z)))/(1-z)^t: Gser := simplify(series(G, z = 0, 17)): for n from 0 to 14 do P[n] := sort(factorial(n)*coeff(Gser, z, n)) end do: for n from 0 to 14 do seq(coeff(P[n], t, k), k = 0 .. floor((1/4)*n)) end do; # yields sequence in triangular form
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