A258579 Triangle read by rows: T(n,k) = number of partial idempotent mappings (of an n-chain) with (right) waist exactly k.

1, 1, 1, 1, 2, 3, 1, 4, 7, 11, 1, 8, 17, 30, 48, 1, 16, 43, 86, 150, 241, 1, 32, 113, 258, 492, 846, 1358, 1, 64, 307, 806, 1686, 3108, 5276, 8445, 1, 128, 857, 2610, 6012, 11904, 21392, 35904, 57256, 1, 256, 2443, 8726, 22230, 47376, 90224, 158880, 263976, 419233

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.

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

Sequence in context: A299714 A171083 A364602 * A263757 A021436 A179738

Adjacent sequences: A258576 A258577 A258578 * A258580 A258581 A258582

Keyword

nonn,tabl

Author

Wafa AlNadabi, Jul 04 2015

Status

approved