OFFSET
0,3
COMMENTS
LINKS
Alois P. Heinz, Rows n = 0..250, flattened
FORMULA
E.g.f.: G(t,z) = exp((1-t)(exp(z)-1))/(1-z)^t.
The 4-variate e.g.f. H(u,v,w,z) of the permutations of {1,2,...,n} with respect to size (marked by z), number of fixed points (marked by u), number of increasing cycles of length >=2 (marked by v), and number of nonincreasing cycles (marked by w) is given by H(u,v,w,z)=exp(uz+v(exp(z)-1-z)+w(1-exp(z))/(1-z)^w. Remark: the nonincreasing cycles are necessarily of length >=3. We have: G(t,z) = H(1,1,t,z).
EXAMPLE
T(3,0)=5 because we have (1)(2)(3), (1)(23), (12)(3), (13)(2), and (123).
T(3,1)=1 because we have (132).
T(4,1)=9 because we have (1)(243), (1432), (142)(3), (132)(4), (1342), (1423), (1243), (143)(2), and (1324).
Triangle starts:
1;
1;
2;
5, 1;
15, 9;
52, 68;
203, 507, 10;
MAPLE
G := exp((1-t)*(exp(z)-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:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(
(1+x*(i!-1))*b(n-i-1)*binomial(n-1, i), i=0..n-1)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..15); # Alois P. Heinz, Sep 26 2016
MATHEMATICA
b[n_] := b[n] = Expand[If[n==0, 1, Sum[(1+x*(i!-1))*b[n-i-1]*Binomial[n-1, i], {i, 0, n-1}]]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Nov 28 2016 after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Feb 26 2011
STATUS
approved