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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)|).

%I A145033 #12 Feb 26 2020 09:05:47
%S A145033 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 A145033 13,56,119,140,91,28,1,1,15,76,196,294,266,140,36,1,1,17,99,300,546,
%U A145033 630,462,204,45,1,1,19,125,435,930,1302,1218,750,285,55,1
%N 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)|).
%C A145033 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 A145033 The matrix inverse starts:
%C A145033 1;
%C A145033 -1,1;
%C A145033 2,-3,1;
%C A145033 -8,13,-6,1;
%C A145033 58,-95,46,-10,1;
%C A145033 -672,1101,-535,120,-15,1;
%C A145033 11374,-18635,9056,-2035,260,-21,1; - _R. J. Mathar_, Mar 29 2013
%H A145033 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 A145033 T(n,k) = C(n,k)*((n-k)*(k+1)+1)/(n-k+1), (n>=k>=0).
%e A145033 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 A145033    1;
%e A145033    1,   1;
%e A145033    1,   3,   1;
%e A145033    1,   5,   6,   1;
%e A145033    1,   7,  14,  10,   1;
%e A145033    1,   9,  25,  30,  15,   1;
%e A145033    1,  11,  39,  65,  55,  21,   1;
%e A145033    1,  13,  56, 119, 140,  91,  28,   1;
%e A145033    1,  15,  76, 196, 294, 266, 140,  36,   1;
%e A145033    1,  17,  99, 300, 546, 630, 462, 204,  45,   1;
%e A145033    1,  19, 125, 435, 930,1302,1218, 750, 285,  55,   1;
%o A145033 (PARI) T(n,k) = binomial(n,k)*((n-k)*(k+1)+1)/(n-k+1);
%o A145033 tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Apr 23 2018
%Y A145033 Row sums of T(n, k) is A005183.
%K A145033 nonn,tabl
%O A145033 0,5
%A A145033 _Abdullahi Umar_, Sep 30 2008
%E A145033 More terms from _Jinyuan Wang_, Feb 26 2020