

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, 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
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OFFSET

0,4


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.


REFERENCES

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


LINKS

Table of n, a(n) for n=0..54.


FORMULA

T(n,k) = n!/k! * Stirling2(n,nk).
T(n,0) = n!.
T(n,k) = A055302(n,k)*(nk) + 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..nk}(1)^j*binomial(nk,j)*(nkj)^n.  Geoffrey Critzer, Aug 20 2013


EXAMPLE

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;


MATHEMATICA

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


CROSSREFS

Cf. A055314.
Sequence in context: A323777 A292317 A285675 * A168615 A174104 A296492
Adjacent sequences: A219856 A219857 A219858 * A219860 A219861 A219862


KEYWORD

nonn,tabl


AUTHOR

Geoffrey Critzer, Dec 01 2012


STATUS

approved



