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A258579 Triangle read by rows: T(n,k) = number of partial idempotent mappings (of an n-chain) with (right) waist exactly k. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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: A189187 A299714 A171083 * A263757 A021436 A179738

Adjacent sequences:  A258576 A258577 A258578 * A258580 A258581 A258582

KEYWORD

nonn,tabl

AUTHOR

Wafa AlNadabi, Jul 04 2015

STATUS

approved

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Last modified January 19 19:49 EST 2019. Contains 319309 sequences. (Running on oeis4.)