OFFSET
0,2
REFERENCES
F. AlKharosi, W. AlNadabi and A. Umar, "Combinatorial results for idempotents in full and partial transformation semigroups", (submitted).
FORMULA
T(n,k) = binomial(n,k)Sum_{r=k}^n binomial(n-k,r-k)Sum_{j=0}^k binomial(k,j)S(k-j,j)j!, where S (x,y) is the Stirling numbers of the second kind, which gives the number of ways to partition x into y nonempty subsets.
EXAMPLE
T (3,2) = 12 because there are exactly 12 partial idempotent mappings (of a 3-chain) with collapse exactly 2, namely: (123-->113), (123-->121), (123-->122), (123-->223), (123-->133), (123--> 323), (12-->11), (12-->22), (23-->22), (23-->33), (13-->11), (13-->33).
Triangle starts:
1;
2,0;
4,0,2;
8,0,12,3;
16,0,48,24,16;
...
PROG
(PARI) tabl(nn) = {for (n=0, nn, for (k=0, n, print1(binomial(n, k)*sum(r=k, n, binomial(n-k, r-k)*sum(j=0, k, binomial(k, j)*stirling(k-j, j, 2)*j!)), ", "); ); print(); ); } \\ Michel Marcus, Jul 15 2015
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Wafa AlNadabi, Jul 04 2015
EXTENSIONS
More terms from Michel Marcus, Jul 15 2015
STATUS
approved