This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A111776 #8 Dec 29 2023 13:16:55 %S A111776 1,1,1,1,2,3,1,4,6,10,1,8,12,20,35,1,16,24,40,70,125,1,32,48,80,140, %T A111776 250,450,1,64,96,160,280,500,900,1625 %N A111776 Triangle read by rows: number of idempotent order-preserving partial transformations (of an n-element chain) of waist k (waist(alpha) = max(Im(alpha))). %C A111776 G(n, n) is A081567(n - 1) %D A111776 Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations. Journal of Algebra 278, (2004), 342-359. %F A111776 G(n,k)= (2^(n-k))*G(n,n)=(2^(n-k))*A081567(n-1), G(0,0) = 1 %e A111776 G(3,2) = 6 because there are exactly 6 idempotent order-preserving partial transformations (on a 3-element chain) of waist 2, namely: (2)->(2), (1,2)->(1,2), (1,2)->(2,2),(1,3)->(3,3), (2,3)->(2,2), (2,3)->(3,3) - the mappings are coordinate-wise %Y A111776 Cf. A081567. %K A111776 nonn,tabl %O A111776 0,5 %A A111776 _Abdullahi Umar_, Aug 25 2008