OFFSET
1,3
REFERENCES
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving full transformations. Semigroup Forum 72, (2006), 51-62.
LINKS
Luigi Santocanale, On discrete idempotent paths, arXiv:1906.05590 [math.LO], 2019.
FORMULA
T(n,k) = Sum_{j=0..k} C(k+j-1, k-1-j) for all n >= k. [Corrected by Georg Fischer, Jul 30 2023]
EXAMPLE
T(4,3) = 5 because there are exactly 5 idempotent order-preserving full transformations (on a 4-element chain) of waist 3, namely: the five possible ordered images (1,1,3,3), (1,2,3,3), (1,3,3,3), (2,2,3,3), (3,3,3,3) of (1,2,3,4).
1;
1, 2;
1, 2, 5;
1, 2, 5, 13;
1, 2, 5, 13, 34;
1, 2, 5, 13, 34, 89;
1, 2, 5, 13, 34, 89, 233;
1, 2, 5, 13, 34, 89, 233, 610;
...
MAPLE
seq(print(seq(add(binomial(k+j-1, k-1-j), j=0..k), k=1..n)), n=1..12); # Georg Fischer, Jul 30 2023
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Abdullahi Umar, Sep 15 2008
EXTENSIONS
a(45) and following terms corrected by Georg Fischer, Jul 30 2023
STATUS
approved