A186361 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k cycles that are not up-down. A cycle (b(1), b(2), ...) is said to be up-down if, when written with its smallest element in the first position, it satisfies b(1)<b(2)>b(3)<... .

1, 1, 2, 5, 1, 16, 8, 61, 59, 272, 438, 10, 1385, 3445, 210, 7936, 29080, 3304, 50521, 264871, 47208, 280, 353792, 2605002, 658806, 11200, 2702765, 27634817, 9275838, 303380, 22368256, 315591124, 134010580, 7016240, 15400, 199360981, 3870632947, 2005021876, 151003996, 1001000

Offset

0,3

Comments

Row n contains 1 + floor(n/3) entries.

Sum of entries in row n is n!.

T(n,0)=A000111(n+1) (the Euler or up-down numbers).

Sum(k*T(n,k),k>=0) = A186362(n).

Links

Alois P. Heinz, Rows n = 0..200, flattened

E. Deutsch and S. Elizalde, Cycle up-down permutations, arXiv:0909.5199v1 [math.CO].

Formula

E.g.f.=(1-sin z)^{s-1}/(1-z)^s.

The trivariate e.g.f. H(t,s,z) of the permutations of {1,2,...,n} with respect to size (marked by z), number of up-down cycles (marked by t), and number of cycles that are not up-down (marked by s) is given by H(t,s,z)=(1-sin z)^{s-t}/(1-z)^s.

Example

T(3,1)=1 because we have (123).
T(4,1)=8 because we have (1432), (1)(234), (1342), (1243), (123)(4), (1234), (124)(3), and (134)(2).
Triangle starts:
    1;
    1;
    2;
    5,   1;
   16,   8;
   61,  59;
  272, 438, 10;
  ...

Maple

G := (1-sin(z))^(t-1)/(1-z)^t: Gser := simplify(series(G, z = 0, 16)): for n from 0 to 13 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: for n from 0 to 13 do seq(coeff(P[n], t, j), j = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form
# second Maple program:
g:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(g(o-1+j, u-j), j=1..u))
    end:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(b(n-j)*
      binomial(n-1, j-1)*((j-1)!*x-g(j-1, 0)*(x-1)), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..14);  # Alois P. Heinz, Apr 15 2017

Mathematica

g[u_, o_] := g[u, o] = If[u + o == 0, 1, Sum[g[o-1+j, u-j], {j, 1, u}]];
b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n - j]*Binomial[n-1, j-1] * ((j - 1)!*x - g[j - 1, 0]*(x - 1)), {j, 1, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}] ][b[n]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Nov 07 2017, after Alois P. Heinz *)

Crossrefs

Cf. A000111, A186362, A186358.

Sequence in context: A216121 A104546 A121632 * A197365 A121579 A214733

Adjacent sequences: A186358 A186359 A186360 * A186362 A186363 A186364

Keyword

nonn,tabf

Author

Emeric Deutsch, Feb 28 2011

Status

approved