The formula for the number of order-preserving selfmappings of a fence

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“…Those not familiar with generating functions may prefer enumerations where the result is expressed in terms of sums of products of choice functions. The results of [6,7,3] are given in this form. Of course, such an expression is easily obtained from a generating function.…”

The Rank of the Semigroup of All Order-Preserving Transformations on a Finite Fence

“…Order-preserving transformations of (finite) fences were first investigated by Currie and Visentin [5] and by Rutkowski [18]. In [5], by using generating functions, the authors calculate the number of order-preserving transformations of a fence with an even number of elements.…”

“…In [5], by using generating functions, the authors calculate the number of order-preserving transformations of a fence with an even number of elements. On the other hand, an exact formula for the number of such transformations, for any natural number n, was given in [18].…”

GENERATING SETS OF SEMIGROUPS OF PARTIAL TRANSFORMATIONS PRESERVING A ZIG-ZAG ORDER ON $\mathbb{N}$

“…Fences were first studied by J. D. Currie, T. I. Visentin, and A. Rutkowski. The exact numbers for such order-preserving full transformations on an n-element fence, where n is even, have been calculated (with the help of generating functions) in [1] and in [9], the author presented the exact formulas for even as well as for odd n. A minimal generating set as well as the rank of the monoid of all orderpreserving transformations on an n-element fence was given in [3]. Moreover, Dimitrova and Koppitz have investigated the monoid of all order-preserving partial injections on an n-element fence.…”