  We present a bijective algorithm with which an arbitrary permutation
decomposes canonically into elementary blocks which we call families, which are
sets with a specified number of ascents and descents. We show that families,
arranged in an arbitrary order in a sequence, are in bijection with
permutations. The permutation decomposes canonically, by inserting parentheses,
into a tree having as nodes a class of permutations which we call primitive.
Primitive permutations can be assembled from very simple data. The data for the
trees into which a permutation decomposes can be written in a form similar to
the decimal classification of a library. We axiomatize that data. It has a
structure very different from the permutation which it encodes, with shuffles
and pairings instead of reorderings. These structures are similar to the
fundamental processes in quantum field theory. Our main bijective structure
algorithm gives explicit, additive multinomial formulae for the number of
permutations with given sets of elements under and over the diagonal, or with
given ascent and descent values. The multinomial expressions obtained this way
give a new class of bicolored set statistics, between set partitions and set
compositions, called shifted multinomials. These provide for the first time
additive multinomial expressions for Eulerian numbers and derangements, as part
of a sequence of new combinatorial objects. These multinomial expressions
satisfy inductive relations involving only immediate neighbors, similar to the
relations satisfied by the Eulerian numbers.
