A206823 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} with exactly k elements x such that |f^(-1)(x)| = 1; n>=0, 0<=k<=n.

1, 0, 1, 2, 0, 2, 3, 18, 0, 6, 40, 48, 144, 0, 24, 205, 1000, 600, 1200, 0, 120, 2556, 7380, 18000, 7200, 10800, 0, 720, 24409, 125244, 180810, 294000, 88200, 105840, 0, 5040, 347712, 1562176, 4007808, 3857280, 4704000, 1128960, 1128960, 0, 40320

Offset: 0

Author

Geoffrey Critzer, Feb 12 2012

Keywords

• nonn
• tabl

Comments

Row sums = n^n, all functions f:{1,2,...,n}->{1,2,...,n}.

T(n,n)= n!, bijections on {1,2,...,n}.

Examples

			Triangle T(n,k) begins:
    1;
    0      1;
    2      0     2;
    3     18     0      6;
   40     48   144      0    24;
  205   1000   600   1200     0     120;
  ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..140, flattened

Crossrefs

Row sums give: A000312.

Column k=0 gives: A231797.

Cf. A231602.

Programs

Maple

with(combinat): C:= binomial:
b:= proc(t, i, u) option remember; `if`(t=0, 1,
      `if`(i<2, 0, b(t, i-1, u) +add(multinomial(t, t-i*j, i$j)
      *b(t-i*j, i-1, u-j)*u!/(u-j)!/j!, j=1..t/i)))
    end:
T:= (n, k)-> C(n, k)*C(n, k)*k! *b(n-k$2, n-k):
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Nov 13 2013

Mathematica

nn = 8; Prepend[CoefficientList[Table[n! Coefficient[Series[(Exp[x] - x + y x)^n, {x, 0, nn}], x^n], {n, 1, nn}], y], {1}] // Flatten

Formula

E.g.f.: Sum_{k=0..n} T(n,k) * y^k * x^n / n! = (exp(x) - x + y*x)^n.