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

0,8

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

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

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)

Example

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

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

Crossrefs

Cf. A008275, A130534, A132393, A164652.

Sequence in context: A111594 A322549 A349645 * A203951 A323591 A105348

Adjacent sequences: A237993 A237994 A237995 * A237997 A237998 A237999

Keyword

nonn,tabl

Author

Geoffrey Critzer, Feb 16 2014

Status

approved