A219859 Triangular array read by rows: T(n,k) is the number of endofunctions, functions f:{1,2,...,n}->{1,2,...,n}, that have exactly k elements with no preimage; n>=0, 0<=k<=n.

1, 1, 0, 2, 2, 0, 6, 18, 3, 0, 24, 144, 84, 4, 0, 120, 1200, 1500, 300, 5, 0, 720, 10800, 23400, 10800, 930, 6, 0, 5040, 105840, 352800, 294000, 63210, 2646, 7, 0, 40320, 1128960, 5362560, 7056000, 2857680, 324576, 7112, 8, 0, 362880, 13063680, 83825280, 160030080, 105099120, 23496480, 1524600, 18360, 9, 0

Offset: 0

Author

Geoffrey Critzer, Dec 01 2012

Keywords

• nonn
• tabl

Comments

Equivalently, T(n,k) is the number of endofunctions whose functional digraph has exactly k leaves.

Equivalently, T(n,k) is the number of doubly rooted trees with k leaves. Here, a doubly rooted tree is a labeled tree in which two special vertices have been selected and the order of the selection matters. [Bona page 266]

Row sums are n^n.

Examples

			Triangle T(n,k) begins:
    1;
    1,     0;
    2,     2,     0;
    6,    18,     3,     0;
   24,   144,    84,     4,   0;
  120,  1200,  1500,   300,   5, 0;
  720, 10800, 23400, 10800, 930, 6, 0;
  ...
		

References

• M. Bona, Introduction to Enumerative Combinatorics, McGraw Hill, 2007.

Crossrefs

Column k=0-1 give: A000142, A001804.

Row sums give A000312.

T(2n,n) gives A288312.

Cf. A055302, A055314, A090657, A101817, A209290, A344053.

Programs

Mathematica

Table[Table[n!/k!StirlingS2[n,n-k],{k,0,n}],{n,0,8}]//Grid

PARI

row(n) = vector(n+1, k, k--; n!/k! * stirling(n,n-k,2)); \\ Michel Marcus, Jan 24 2022

Formula

T(n,k) = n!/k! * Stirling2(n,n-k).

T(n,0) = n!.

T(n,k) = A055302(n,k)*(n-k) + A055302(n,k+1)*(k+1). The first term (on rhs of this equation) is the number of such functions in which the preimage of f(n) contains more than one element. The second term is the number of such functions in which the preimage of f(n) contains exactly one element.

T(n,k) = binomial(n,k) Sum_{j=0..n-k}(-1)^j*binomial(n-k,j)*(n-k-j)^n. - Geoffrey Critzer, Aug 20 2013

E.g.f.: 1/(1 - (A(x,y) - y*x + x)) where A(x,y) is the e.g.f. for A055302. - Geoffrey Critzer, Jan 24 2022

From Alois P. Heinz, Jan 24 2022: (Start)

Sum_{k=0..n} k * T(n,k) = A209290(n).

Sum_{k=0..n} (-1)^k * T(n,k) = A344053(n). (End)