OFFSET
0,5
REFERENCES
F. AlKharosi, W. AlNadabi and A. Umar, "Combinatorial results for idempotents in full and partial transformation semigroups", (submitted).
LINKS
Haoliang Wang, Robert Simon, The Analysis of Synchronous All-to-All Communication Protocols for Wireless Systems, Q2SWinet'18: Proceedings of the 14th ACM International Symposium on QoS and Security for Wireless and Mobile Networks (2018), 39-48.
FORMULA
T(n,k) = binomial(n,k) * Sum_{m=0..k} binomial(k,m)*m^(k-m).
EXAMPLE
T(3,2) = 9 because there are exactly 9 partial idempotent mappings (of a 3-chain) with breadth exactly 2, namely: (12-->11), (12-->22), (12-->12), (13-->11), (13-->33), (13-->13), (23-->22), (23-->33), (23-->23).
Triangle starts:
1;
1, 1;
1, 2, 3;
1, 3, 9, 10;
1, 4, 18, 40, 41;
...
PROG
(PARI) tabl(nn) = {for (n=0, nn, for (k=0, n, print1(binomial(n, k)*sum(m=0, k, binomial(k, m)*m^(k-m)), ", "); ); 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