A259760 Triangle read by rows: T(n,k) is the number of partial idempotent mappings (of an n-chain) with breadth exactly k.

1, 1, 1, 1, 2, 3, 1, 3, 9, 10, 1, 4, 18, 40, 41, 1, 5, 30, 100, 205, 196, 1, 6, 45, 200, 615, 1176, 1057, 1, 7, 63, 350, 1435, 4116, 7399, 6322, 1, 8, 84, 560, 2870, 10976, 29596, 50576, 41393, 1, 9, 108, 840, 5166, 24696, 88788, 227592, 372537, 293608

Offset

0,5

References

F. AlKharosi, W. AlNadabi and A. Umar, "Combinatorial results for idempotents in full and partial transformation semigroups", (submitted).

Links

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

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

Cf. A000248, A052512.

Row sums give A080108(n+1).

Sequence in context: A135091 A171150 A111589 * A010027 A151880 A108990

Adjacent sequences: A259757 A259758 A259759 * A259761 A259762 A259763

Keyword

nonn,tabl

Author

Wafa AlNadabi, Jul 04 2015

Extensions

More terms from Michel Marcus, Jul 15 2015

Status

approved