Geodesic
Combinatorial results for semigroups of order-decreasing partial transformations
Journal of integer sequences, Tome 7 (2004) no. 3
Let $PC_{n}$ be the semigroup of all decreasing and order-preserving partial transformations of a finite chain. It is shown that |$PC_{n}| = r_{n}$, where $r_{n}$ is the large (or double) Schröder number. Moreover, the total number of idempotents of $PC_{n}$ is shown to be $(3^{n}+1)/2$.
Classification : 20M18, 20M20, 05A10, 05A15
Keywords: semigroup, order-preserving, order-decreasing, partial transformation, full transformation, Catalan number, Fibonacci number, narayana numbers, schr$\ddot $oder numbers 
@article{JIS_2004__7_3_a5,
     author = {Laradji,  A. and Umar,  A.},
     title = {Combinatorial results for semigroups of order-decreasing partial transformations},
     journal = {Journal of integer sequences},
     year = {2004},
     volume = {7},
     number = {3},
     zbl = {1064.05016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2004__7_3_a5/}
}
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Laradji,  A.; Umar,  A. Combinatorial results for semigroups of order-decreasing partial transformations. Journal of integer sequences, Tome 7 (2004) no. 3. http://geodesic.mathdoc.fr/item/JIS_2004__7_3_a5/