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A187247 Triangle read by rows: T(n,k) is the number of permutations of [n] having k cycles with at most 2 alternating runs (it is assumed that the smallest element of the cycle is in the first position), 0<=k<=n. 4

%I

%S 1,0,1,0,1,1,0,2,3,1,2,4,11,6,1,16,18,40,35,10,1,104,142,178,195,85,

%T 15,1,688,1236,1106,1148,665,175,21,1,5116,10832,9300,7728,5173,1820,

%U 322,28,1,44224,98492,91680,63284,42168,18165,4284,546,36,1,438560,964172,974924,627420,378620,181797,53361,9030,870,45,1

%N Triangle read by rows: T(n,k) is the number of permutations of [n] having k cycles with at most 2 alternating runs (it is assumed that the smallest element of the cycle is in the first position), 0<=k<=n.

%C Sum of entries in row n is n!.

%C T(n,0) = A187248(n).

%C Sum(k*T(n,k),k>=0) = A187249(n).

%H Alois P. Heinz, <a href="/A187247/b187247.txt">Rows n = 0..140, flattened</a>

%F E.g.f.: G(t,z) = exp[(1/4)(t-1)(2z - 1 + exp(2z))]/(1-z).

%F The 4-variate g.f. H(u,v,w,z) (exponential with respect z), where u marks number of cycles with 1 alternating run, v marks number of cycles with 2 alternating runs, w marks the number of all cycles, and z marks the size of the permutation, is given by

%F H(u,v,w,z) = exp[(1/4)w((v-1)(exp(2z)+2z)+4(u-v)exp(z)+1-4u+3v)]/(1-z)^w.

%F We have G(t,z) = H(t,t,1,z).

%e T(3,2)=3 because we have (1)(23), (12)(3), and (13)(2).

%e T(4,0)=2 because we have (1423) and (1324).

%e T(4,1)=4 because we have (1234), (1243), (1342), and (1432).

%e Triangle starts:

%e 1;

%e 0,1;

%e 0,1,1;

%e 0,2,3,1;

%e 2,4,11,6,1;

%e 16,18,40,35,10,1;

%p G := exp((1/4)*(t-1)*(2*z-1+exp(2*z)))/(1-z): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 13 do P[n] := sort(expand(factorial(n) * coeff(Gser, z, n))) end do: for n from 0 to 10 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form

%p # second Maple program:

%p b:= proc(n) option remember; expand(

%p `if`(n=0, 1, add(b(n-j)*binomial(n-1, j-1)*

%p `if`(j=1, x, (j-1)!+2^(j-2)*(x-1)), j=1..n)))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):

%p seq(T(n), n=0..12); # _Alois P. Heinz_, Apr 15 2017

%t b[n_] := b[n] = Expand[If[n==0, 1, Sum[b[n-j]*Binomial[n-1, j-1]*If[j==1, x, (j-1)! + 2^(j-2)*(x-1)], {j, 1, n}]]];

%t T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n]];

%t Table[T[n], {n, 0, 12}] // Flatten (* _Jean-Fran├žois Alcover_, May 18 2017, after _Alois P. Heinz_ *)

%Y Cf. A187248, A187249.

%K nonn,tabl

%O 0,8

%A _Emeric Deutsch_, Mar 07 2011

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Last modified January 17 06:12 EST 2022. Contains 350378 sequences. (Running on oeis4.)