OFFSET
0,5
COMMENTS
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
FORMULA
E.g.f.: G(t,z) = exp((t-1)(cosh z - 1))/(1-z).
The 5-variate e.g.f. H(x,y,u,v,z) of permutations with respect to size (marked by z), number of increasing odd cycles (marked by x), number of increasing even cycles (marked by y), number of nonincreasing odd cycles (marked by u), and number of nonincreasing even cycles (marked by v), is given by
H(x,y,u,v,z) = exp(((x-u)sinh z + (y-v)(cosh z - 1))*(1+z)^{(u-v)/2}/(1-z)^{(u+v)/2}.
We have: G(t,z) = H(1,t,1,1,z).
EXAMPLE
T(3,1)=3 because we have (1)(23), (12)(3), and (13)(2).
T(4,2)=3 because we have (12)(34), (13)(24), and (14)(23).
Triangle starts:
1;
1;
1,1;
3,3;
14,7,3;
70,35,15;
MAPLE
g := exp((t-1)*(cosh(z)-1))/(1-z): gser := simplify(series(g, z = 0, 15)): for n from 0 to 12 do P[n] := sort(expand(factorial(n)*coeff(gser, z, n))) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; expand(
`if`(n=0, 1, add(b(n-j)*binomial(n-1, j-1)*
`if`(j::odd, (j-1)!, x+((j-1)!-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 13 2017
MATHEMATICA
b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n-j]*Binomial[n-1, j-1]*If[ OddQ[j], (j-1)!, x+(j-1)!-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, May 03 2017, after Alois P. Heinz *)
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Feb 27 2011
STATUS
approved