A237996 Triangular array read by rows. T(n,k) is the number of even permutations of {1,2,...,n} that have exactly k cycles, n >= 0, 0 <= k <= n.

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 11, 0, 1, 0, 24, 0, 35, 0, 1, 0, 0, 274, 0, 85, 0, 1, 0, 720, 0, 1624, 0, 175, 0, 1, 0, 0, 13068, 0, 6769, 0, 322, 0, 1, 0, 40320, 0, 118124, 0, 22449, 0, 546, 0, 1, 0, 0, 1026576, 0, 723680, 0, 63273, 0, 870, 0, 1

Offset: 0

Author

Geoffrey Critzer, Feb 16 2014

Keywords

• nonn
• tabl

Comments

Row sums = A001710(n).

If a permutation of {1, 2, ..., n} is written as a product of m disjoint cycles (where the fixed points of the permutation are viewed as 1-cycles) then the parity of the permutation is (-1)^(n-m). It is an even permutation if the number of cycles of even length is even (possibly zero), and it is an odd permutation if the number of cycles of even length is odd. - Peter Bala, Jun 25 2024

Examples

			Triangle begins:
  1;
  0,     1;
  0,     0,       1;
  0,     2,       0,      1;
  0,     0,      11,      0,      1;
  0,    24,       0,     35,      0,     1;
  0,     0,     274,      0,     85,     0,     1;
  0,   720,       0,   1624,      0,   175,     0,   1;
  0,     0,   13068,      0,   6769,     0,   322,   0,   1;
  0, 40320,       0, 118124,      0, 22449,     0, 546,   0, 1;
  0,     0, 1026576,      0, 723680,     0, 63273,   0, 870, 0, 1;
  ...
		

References

• J. Riordan, Introduction to Combinatorial Analysis, Wiley, 1958, page 87, problem # 20.

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Wikipedia, Parity of a permutaion

Crossrefs

Cf. A008275, A130534, A132393, A164652.

Programs

Maple

with(combinat):
b:= proc(n, i, t) option remember; expand(`if`(n=0, t, `if`(i<1,
      0, add(x^j*multinomial(n, n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j,
      i-1, irem(t+`if`(irem(i,2)=0, j, 0), 2)), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 09 2015
# Alternative:
A132393 := (n, k) -> abs(Stirling1(n, k)):
T := (n, k) -> ifelse((n::even and k::even) or (n::odd and k::odd), A132393(n, k),
0): seq(seq(T(n, k), k = 0..n), n = 0..9);  # Peter Luschny, Jun 26 2024

Mathematica

nn=11;a=Log[((1+x)/(1-x))^(1/2)];b=Log[1/(1-x^2)^(1/2)];Table[Take[(Range[0,nn]!CoefficientList[Series[Exp[y a]Cosh[y b] ,{x,0,nn}],{x,y}])[[n]],n],{n,1,nn}]//Grid

Formula

E.g.f.: exp(y*A(x))*cosh(y*B(x)) where A(x) = log(((1 + x)/(1 - x))^(1/2)) and B(x) = log((1/(1-x^2))^(1/2)).

From Peter Bala, Jun 25 2024: (Start)

If n and k are both even or both odd, then T(n, k) is equal to the Stirling cycle number |s(n, k)| = A132393(n, k), and 0 otherwise.

n-th row polynomial R(n, x) = (1/2)*( x*(x + 1)*...*(x + n + 1) + x*(x - 1)*...*(x - n - 1) ).

For n >= 1, the zeros of R(n, x) are purely imaginary. (End)