OFFSET
0,5
REFERENCES
F. AlKharosi, W. AlNadabi and A. Umar, "Combinatorial results for idempotents in full and partial transformation semigroups", (submitted).
FORMULA
T(n,k) = Sum_{m=0..k} binomial(k-1,m-1) * (m+1)^(n-m).
EXAMPLE
T(3,2) = 7 because there are exactly 7 partial idempotent mappings (of a 3-chain) with right waist exactly 2, namely: (123-->222), (123-->122), (123-->121), (12-->22), (12-->12), (23-->22), (2-->2).
Triangle starts:
1;
1,1;
1,2,3;
1,4,7,11;
1,8,17,30,48;
...
PROG
(PARI) mybinom(x, y) = if ((x==-1) && (y==-1), 1, binomial(x, y));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(sum(m=0, k, mybinom(k-1, m-1) * (m+1)^(n-m)), ", "); ); print(); ); } \\ Michel Marcus, Jul 15 2015
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Wafa AlNadabi, Jul 04 2015
STATUS
approved