Geodesic
Semiorders and Riordan numbers
Journal of integer sequences, Tome 10 (2007) no. 7
In this paper, we define a class of semiorders (or unit interval orders) that arose in the context of polyhedral combinatorics. In the first section of the paper, we will present a pure counting argument equating the number of these interesting (connected and irredundant) semiorders on $n+1$ elements with the $n$th Riordan number. In the second section, we will make explicit the relationship between the interesting semiorders and a special class of Motzkin paths, namely, those Motzkin paths without horizontal steps of height 0, which are known to be counted by the Riordan numbers.
Classification : 05A15, 06A07
Keywords: riordan numbers, interval orders, semiorders, lattice paths 
@article{JIS_2007__10_7_a7,
     author = {Balof,  Barry and Menashe,  Jacob},
     title = {Semiorders and {Riordan} numbers},
     journal = {Journal of integer sequences},
     year = {2007},
     volume = {10},
     number = {7},
     zbl = {1141.05304},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2007__10_7_a7/}
}
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Balof,  Barry; Menashe,  Jacob. Semiorders and Riordan numbers. Journal of integer sequences, Tome 10 (2007) no. 7. http://geodesic.mathdoc.fr/item/JIS_2007__10_7_a7/