A214851 Irregular triangular array read by rows. T(n,k) is the number of n-permutations that have exactly k square roots. n >= 1, 0 <= k <= A000085(n).

0, 1, 1, 0, 1, 3, 2, 0, 0, 1, 12, 8, 3, 0, 0, 0, 0, 0, 0, 0, 1, 60, 24, 35, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 450, 184, 0, 0, 85, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Geoffrey Critzer, Mar 08 2013

Keywords

• nonn
• tabf

Comments

Row sums = n!.

Sum_{k=1...A000085(n)} T(n,k)*k = n!.

Sum_{k=1...A000085(n)} T(n,k) = A003483(n).

Column k=0 is n! - A003483(n).

Examples

			0, 1,
1, 0, 1,
3, 2, 0, 0, 1,
12, 8, 3, 0, 0, 0, 0, 0, 0, 0, 1,
60, 24, 35, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
450, 184, 0, 0, 85, 0,0,0,...,1 where the 1 is in column k=76.
T(5,2)= 35 because we have 20 5-permutations of the type (1,2,3)(4)(5) and 15 of the type (1,2)(3,4)(5).  These have 2 square roots:(1,3,2)(4)(5),(1,3,2)(4,5) and (1,3,2,4)(5),(3,1,4,2)(5) respectively.
		

Crossrefs

Cf. A214849 (column k=1), A214854 (column k=2).

Programs

Mathematica

(* Warning: the code is very inefficient, it takes about one minute to run on a laptop computer. *) a={1,2,4,10,26}; Table[Distribution[Distribution[Table[MultiplicationTable[Permutations[m], Permute[#1,#2]&][[n]][[n]], {n,1,m!}], Range[1,m!]], Range[0,a[[m]]]], {m,1,5}] //Grid