%I #12 Feb 26 2020 09:05:47

%S 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,

%T 13,56,119,140,91,28,1,1,15,76,196,294,266,140,36,1,1,17,99,300,546,

%U 630,462,204,45,1,1,19,125,435,930,1302,1218,750,285,55,1

%N 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)|).

%C T(n,k) is also the rank of the semigroup of order-decreasing partial one-one transformations (of an n-chain) of height <= k.

%C The matrix inverse starts:

%C 1;

%C -1,1;

%C 2,-3,1;

%C -8,13,-6,1;

%C 58,-95,46,-10,1;

%C -672,1101,-535,120,-15,1;

%C 11374,-18635,9056,-2035,260,-21,1; - _R. J. Mathar_, Mar 29 2013

%H A. Umar, <a href="https://www.emis.de/journals/PM/53f1/pm53f102.pdf">On the ranks of certain finite semigroups of order-decreasing transformations</a> Portugaliae Math. 53, (1996), 23-34.

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

%e 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).

%e 1;

%e 1, 1;

%e 1, 3, 1;

%e 1, 5, 6, 1;

%e 1, 7, 14, 10, 1;

%e 1, 9, 25, 30, 15, 1;

%e 1, 11, 39, 65, 55, 21, 1;

%e 1, 13, 56, 119, 140, 91, 28, 1;

%e 1, 15, 76, 196, 294, 266, 140, 36, 1;

%e 1, 17, 99, 300, 546, 630, 462, 204, 45, 1;

%e 1, 19, 125, 435, 930,1302,1218, 750, 285, 55, 1;

%o (PARI) T(n,k) = binomial(n,k)*((n-k)*(k+1)+1)/(n-k+1);

%o tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Apr 23 2018

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

%K nonn,tabl

%O 0,5

%A _Abdullahi Umar_, Sep 30 2008

%E More terms from _Jinyuan Wang_, Feb 26 2020