A145033 T(n,k) is the number of amenable quasi-idempotent order-decreasing partial one-one transformations (of an n-chain) of height k (height(alpha) = |Im(alpha)|).

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

Offset: 0

Author

Abdullahi Umar, Sep 30 2008

Keywords

• nonn
• tabl

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

Examples

			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;
		

Links

• A. Umar, On the ranks of certain finite semigroups of order-decreasing transformations Portugaliae Math. 53, (1996), 23-34.

Crossrefs

Row sums of T(n, k) is A005183.

Programs

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

Formula

T(n,k) = C(n,k)*((n-k)*(k+1)+1)/(n-k+1), (n>=k>=0).

Extensions

More terms from Jinyuan Wang, Feb 26 2020