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A186757
Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k increasing cycles of length >=2 (0<=k<= n/2). A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1) < b(2) < b(3) < ... .
7
1, 1, 1, 1, 2, 4, 10, 11, 3, 59, 36, 25, 363, 212, 130, 15, 2491, 1688, 651, 210, 19661, 14317, 4487, 1750, 105, 176536, 129076, 42435, 12628, 2205, 1767540, 1277159, 451626, 104755, 26775, 945, 19460671, 13974236, 5068723, 1120570, 264880, 27720
OFFSET
0,5
COMMENTS
Row n contains 1 + floor(n/2) entries.
Sum of entries in row n is n!.
T(n,0) = A186758(n).
Sum_{k>=0} k*T(n,k) = A056542(n).
LINKS
FORMULA
E.g.f.: G(t,z) = exp((t-1)(exp(z)-1-z))/(1-z).
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,t,1,z).
EXAMPLE
T(3,0)=2 because we have (1)(2)(3) and (132).
T(4,2)=3 because we have (13)(24), (12)(34), and (14)(23).
Triangle starts:
1;
1;
1, 1;
2, 4;
10, 11, 3;
59, 36, 25;
363, 212, 130, 15;
MAPLE
b:= proc(n) option remember; expand(
`if`(n=0, 1, add(b(n-i)*binomial(n-1, i-1)*
`if`(i>1, (x+(i-1)!-1), 1), i=1..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..12); # Alois P. Heinz, Mar 19 2017
MATHEMATICA
b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n-i]*Binomial[n-1, i-1]*If[i > 1, (x + (i - 1)! - 1), 1], {i, 1, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 03 2017, after Alois P. Heinz *)
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Feb 26 2011
STATUS
approved