OFFSET
0,5
COMMENTS
T(n,k) is also the rank of the semigroup of order-decreasing partial one-one transformations (of an n-chain) of height <= k.
The matrix inverse starts:
1;
-1,1;
2,-3,1;
-8,13,-6,1;
58,-95,46,-10,1;
-672,1101,-535,120,-15,1;
11374,-18635,9056,-2035,260,-21,1; - R. J. Mathar, Mar 29 2013
LINKS
A. Umar, On the ranks of certain finite semigroups of order-decreasing transformations Portugaliae Math. 53, (1996), 23-34.
FORMULA
T(n,k) = C(n,k)*((n-k)*(k+1)+1)/(n-k+1), (n>=k>=0).
EXAMPLE
T(3,2) = 6 because there are exactly 6 amenable quasi-idempotent order-decreasing partial one-one transformations (on a 3- chain) of height 2, namely: (1,2)->(1,2), (1,3)->(1,2), (1,3)->(1,3), (2,3)->(1,3), (2,3)->(2,1), (2,3)->(2,3).
1;
1, 1;
1, 3, 1;
1, 5, 6, 1;
1, 7, 14, 10, 1;
1, 9, 25, 30, 15, 1;
1, 11, 39, 65, 55, 21, 1;
1, 13, 56, 119, 140, 91, 28, 1;
1, 15, 76, 196, 294, 266, 140, 36, 1;
1, 17, 99, 300, 546, 630, 462, 204, 45, 1;
1, 19, 125, 435, 930,1302,1218, 750, 285, 55, 1;
PROG
(PARI) T(n, k) = binomial(n, k)*((n-k)*(k+1)+1)/(n-k+1);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Apr 23 2018
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Abdullahi Umar, Sep 30 2008
EXTENSIONS
More terms from Jinyuan Wang, Feb 26 2020
STATUS
approved