A161119 Triangle read by rows: T(n,k) is the number of fixed-point-free involutions of {1,2,...,2n} having k cycles with entries of opposite parities (0 <= k <= n).

1, 0, 1, 1, 0, 2, 0, 9, 0, 6, 9, 0, 72, 0, 24, 0, 225, 0, 600, 0, 120, 225, 0, 4050, 0, 5400, 0, 720, 0, 11025, 0, 66150, 0, 52920, 0, 5040, 11025, 0, 352800, 0, 1058400, 0, 564480, 0, 40320, 0, 893025, 0, 9525600, 0, 17146080, 0, 6531840, 0, 362880, 893025, 0, 44651250, 0, 238140000, 0, 285768000, 0, 81648000, 0, 3628800

Offset: 0

Author

Emeric Deutsch, Jun 02 2009

Keywords

• nonn
• tabl

Comments

T(n,k) is the number of basis elements in the order-n Brauer algebra that have propagation number k. - John M. Campbell, Dec 08 2021

Examples

			T(3,1)=9 because we have (12)(35)(46), (14)(26)(35), (16)(24)(35), (23)(15)(46), (25)(13)(46), (34)(15)(26), (36)(15)(24), (45)(13)(26), (56)(13)(24).
Triangle starts:
  1;
  0,  1;
  1,  0,  2;
  0,  9,  0,  6;
  9,  0, 72,  0, 24;
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Crossrefs

Cf. A000142, A001147, A001818, A161120, A161121, A161122.

Programs

Maple

T := proc (n, k) if `mod`(n-k, 2) = 1 then 0 else binomial(n, k)^2*factorial(k)*(product(2*j-1, j = 1 .. (1/2)*n-(1/2)*k))^2 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form

PARI

dfo(n) = if (n<0, (-1)^n/dfo(-n), (2*n)! / n! / 2^n); \\ A001147
T(n,k) = if ((n-k)%2, 0, k!*binomial(n,k)^2*dfo((n-k)/2)^2);
row(n) = vector(n+1, k, T(n,k-1)) \\ Michel Marcus, Dec 09 2021

Formula

T(n,k) = k!*binomial(n,k)^2*(n-k-1)!!^2 if n-k is even; T(n,k) = 0 if n-k is odd.

Sum of row n = (2n-1)!! = A001147(n).

T(n,n) = n! = A000142(n).

T(2n,0) = A001818(n).

Sum_{k>=0} k*T(n,k) = n^2*(2n-3)!! = A161120(n).