A381529 T(n,k) is the number of permutations of [n] having exactly k pairs of integers i=0, 0<=k<=A125811(n)-1, read by rows.

1, 1, 2, 5, 1, 15, 5, 4, 54, 21, 24, 16, 5, 235, 89, 118, 112, 101, 35, 28, 2, 1237, 408, 577, 633, 719, 585, 402, 239, 167, 59, 14, 7790, 2106, 3023, 3529, 4410, 4463, 4600, 3012, 2789, 1933, 1438, 629, 442, 122, 34, 57581, 12529, 17693, 20980, 27208, 30064, 35359, 33332, 28137, 24970, 22850, 17148, 14272, 8645, 5639, 3684, 1809, 664, 282, 34

Offset: 0

Author

Alois P. Heinz, Feb 26 2025

Keywords

• nonn
• tabf

Examples

			T(4,0) = 15: (1)(2)(3)(4), (1,2)(3)(4), (1)(2,3)(4), (1)(2)(3,4), (1,2)(3,4), (1,2,3)(4), (1,3,2)(4), (1)(2,3,4), (1)(2,4,3), (1,2,3,4), (1,2,4,3), (1,3,2,4), (1,3,4,2), (1,4,2,3), (1,4,3,2).
T(4,1) = 5: (1)(2,4)(3), (1,2,4)(3), (1,4,2)(3), (1,3)(2)(4), (1,3)(2,4).
T(4,2) = 4: (1,4)(2)(3), (1,4)(2,3), (1,3,4)(2), (1,4,3)(2).
Triangle T(n,k) begins:
     1;
     1;
     2;
     5,   1;
    15,   5,   4;
    54,  21,  24,  16,   5;
   235,  89, 118, 112, 101,  35,  28,   2;
  1237, 408, 577, 633, 719, 585, 402, 239, 167, 59, 14;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..55, flattened
• Wikipedia, Inversion
• Wikipedia, Permutation

Crossrefs

Columns k=0-1 give: A051295, A381539.

Row sums give A000142.

Row lengths give A125811.

Last elements of rows give A381531.

Main diagonal gives A381545.

Cf. A008302, A125810 (similar for set partitions), A126673, A381299 (similar for ordered set partitions).

Programs

Maple

b:= proc(o, u, t) option remember; expand(`if`(u+o=0, max(0, t-1)!,
     `if`(t>0, b(u+o, 0$2)*(t-1)!, 0)+add(x^(u+j-1)*
        b(o-j, u+j-1, t+1), j=`if`(t=0, 1, 1..o))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..10);

Formula

Sum_{k>=1} k * T(n,k) = A126673(n)/2.