Recall that I_n is the inverse monoid of partial

bijections of n where n= 1,2,...,n .

We show that [THETA] and [PHI] are mutually inverse

bijections. Due to symmetry, it suffices to show that [THETA]([PHI](I)) = I for any I [member of] Id(S).

Let it be A = (P, L, I) an affine plane and S = {[psi]: P [right arrow] P| where [psi]--is bijection} set of

bijections to set points P on yourself.

We use extensively the

bijections [[psi].sub.n] and [[??].sub.n] between [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the

bijections [[PHI].sub.n] and [[??].sub.n] between [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], studied in [4] (see Appendix for more information).

In fact, for each group G, there is a universal inverse semigroup S(G), nowadays known as Exel's semigroup, which associates to each partial action of G on a set (topological space) X, a morphism of semigroups between S(G) and the inverse semigroup of partially defined

bijections (homeomorphisms) in X.

For the first one, through

bijections, we give relations between the number of corners in permutation tableaux, alternative tableaux and tree-like tableaux.

Also, we present constructive

bijections between the set of Motzkin paths of length n - 1 and the sets of irreducible permutations of length n (respectively fixed point free irreducible involutions of length 2n) avoiding a pattern [alpha] for [alpha] [member of] {132, 213, 321}.

Our general approach is to merge several steps of each round function of SHARK into table lookups, blending by randomly generated mixing

bijections. We use techniques from [10, 12] to obtain the obfuscated implementation.

Specially, if we have

bijections a:V {1,2,...,p} or :E {1,2,...,q} then the labelings are called respectively a vertex labeling or a edge labeling.

BORED with separating sets of numbers into piles of

bijections and Cayleigh tables, this weekend, I took to rummaging through a seldom opened cupboard.

The set SYM(G,*) = SYM(G) of all

bijections in G forms a group called the permutation (symmetric) group of G.

Morphisms between A-weighted sets are weight preserving

bijections. The cardinality [[absolute value of X].sub.f] of a weighted set (X, f) is given by