  Recently, in papers by Knutson, Tao and Woodward, Henriques and Kamnitzer,
Pak and Vallejo have been constructed several interesting bijections of
associativity and commutativity. In the first two papers bijections relate
special sets of discretely concave functions (hives) on triangular grids and
the octahedron recurrence plays the key role for these bijections. Pak and
Vallejo related special sets of Young tableaux and constructions of these
bijections based on standard algorithms in this theory, jeu de taquen,
Schutzenberger involution, tableaux switching, etc. In this paper we
investigate these constructions from the third point of view, combinatorics of
arrays, theory worked out recently by the authors. Arrays naturally related as
well to functions on the lattice of integers as to Young tableaux. In the
tensor category of arrays, the bijections of associativity and commutativity
arise naturally. We establish coincidence of our bijections with that defined
in the first two papers and in the integer-valued set-up with the bijection in
the third paper (that is, in particular, a solution of Conjecture 1 by Pak and
Vallejo). In order to relate different approaches and to reveal combinatorics
of the octahedron recurrence, we, first, show that the octahedron recurrence
agrees with discrete convexity and, second, we construct another bijection
using the octahedron recurrence, the functional form of the RSK correspondence.
