|
|
A259759
|
|
Triangle read by rows: T(n,k) = number of partial idempotent mappings (of an n-chain) with collapse exactly k.
|
|
0
|
|
|
1, 2, 0, 4, 0, 2, 8, 0, 12, 3, 16, 0, 48, 24, 16, 32, 0, 160, 120, 160, 65, 64, 0, 480, 480, 960, 780, 336, 128, 0, 1344, 1680, 4480, 5460, 4704, 1897, 256, 0, 3584, 5376, 17920, 29120, 37632, 30352, 11824, 512, 0, 9216, 16128, 64512, 131040, 225792, 273168, 212832, 80145
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
REFERENCES
|
F. AlKharosi, W. AlNadabi and A. Umar, "Combinatorial results for idempotents in full and partial transformation semigroups", (submitted).
|
|
LINKS
|
|
|
FORMULA
|
T(n,k) = binomial(n,k)Sum_{r=k}^n binomial(n-k,r-k)Sum_{j=0}^k binomial(k,j)S(k-j,j)j!, where S (x,y) is the Stirling numbers of the second kind, which gives the number of ways to partition x into y nonempty subsets.
|
|
EXAMPLE
|
T (3,2) = 12 because there are exactly 12 partial idempotent mappings (of a 3-chain) with collapse exactly 2, namely: (123-->113), (123-->121), (123-->122), (123-->223), (123-->133), (123--> 323), (12-->11), (12-->22), (23-->22), (23-->33), (13-->11), (13-->33).
Triangle starts:
1;
2,0;
4,0,2;
8,0,12,3;
16,0,48,24,16;
...
|
|
PROG
|
(PARI) tabl(nn) = {for (n=0, nn, for (k=0, n, print1(binomial(n, k)*sum(r=k, n, binomial(n-k, r-k)*sum(j=0, k, binomial(k, j)*stirling(k-j, j, 2)*j!)), ", "); ); print(); ); } \\ Michel Marcus, Jul 15 2015
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|