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%I #4 Mar 31 2012 14:40:12
%S 1,1,1,1,2,2,1,4,4,5,1,8,8,10,14,1,16,16,20,28,41,1,32,32,40,56,82,
%T 122,1,64,64,80,112,164,244,365,1,128,128,160,224,328,488,730,1094,1,
%U 256,256,320,448,656,976,1460,2188,3281,1,512,512,640,896,1312,1952,2920,4376,6562,9842
%N T(n,k) is the number of idempotent order-decreasing and order-preserving partial transformations (of an n-chain) of waist k (waist(alpha) = max(Im(alpha))).
%H Laradji, A. and Umar, A., <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Umar/um.html">Combinatorial Results for Semigroups of Order-Decreasing Partial Transformations </a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.8. [From _Abdullahi Umar_, Oct 07 2008]
%F T(n,k)=2^(n-k-1)(1+3^(k-1)), k>0.
%e T(3,2) = 4 because there are exactly 4 idempotent order-decreasing and order-preserving partial transformations (of a 3-chain) of waist 2, namely: 2->2, (1,2)->(1,2), (2,3)->(2,2), (1,2,3)->(1,2,2).
%e 1;
%e 1,1;
%e 1,2,2;
%e 1,4,4,5;
%e 1,8,8,10,14;
%e 1,16,16,20,28,41;
%e 1,32,32,40,56,82,122;
%e 1,64,64,80,112,164,244,365;
%e 1,128,128,160,224,328,488,730,1094;
%e 1,256,256,320,448,656,976,1460,2188,3281;
%e 1,512,512,640,896,1312,1952,2920,4376,6562,9842;
%Y Sum of rows of T(n, k) is A007051 and T(n, k)=2^(n-k)A007051(k-1) (n>=k>=1)
%K nonn,tabl
%O 0,5
%A _Abdullahi Umar_, Sep 30 2008
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