A145036 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))).

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, 122, 1, 64, 64, 80, 112, 164, 244, 365, 1, 128, 128, 160, 224, 328, 488, 730, 1094, 1, 256, 256, 320, 448, 656, 976, 1460, 2188, 3281, 1, 512, 512, 640, 896, 1312, 1952, 2920, 4376, 6562, 9842

Offset

0,5

Links

Table of n, a(n) for n=0..65.

Laradji, A. and Umar, A., Combinatorial Results for Semigroups of Order-Decreasing Partial Transformations , Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.8. [From Abdullahi Umar, Oct 07 2008]

Formula

T(n,k)=2^(n-k-1)(1+3^(k-1)), k>0.

Example

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).
1;
1,1;
1,2,2;
1,4,4,5;
1,8,8,10,14;
1,16,16,20,28,41;
1,32,32,40,56,82,122;
1,64,64,80,112,164,244,365;
1,128,128,160,224,328,488,730,1094;
1,256,256,320,448,656,976,1460,2188,3281;
1,512,512,640,896,1312,1952,2920,4376,6562,9842;

Crossrefs

Sum of rows of T(n, k) is A007051 and T(n, k)=2^(n-k)A007051(k-1) (n>=k>=1)

Sequence in context: A261357 A238870 A213946 * A341146 A355276 A272888

Adjacent sequences: A145033 A145034 A145035 * A145037 A145038 A145039

Keyword

nonn,tabl

Author

Abdullahi Umar, Sep 30 2008

Status

approved