We study various combinatorial aspects of concrete algebras
satisfying nontrivial polynomial identities (PI-algebras). In
particular for the near-triangular matrix algebras we found a
basis of their polynomial identities. The general approach to
investigate the growth of identities for PI-algebras is to
consider their codimension sequences. As of yet the knowledge
about codimensions is less than satisfactory. There are only a
few concrete algebras with known sequences of codimensions. Here
we asymptotically evaluate the growth of identities for
near-triangular matrix algebras.
It is shown that if an ideal of identities of a certain
PI-algebra asymptotically contains all
proper identities then this
algebra has a polynomial growth of codimensions.
An algebra formed by upper triangular matrices with equal
elements on the main diagonal is investigated. It is proved that
the ideal of identities of this algebra gives an example of
ideal which asymptotically contains all proper identities. The
codimension sequence for this algebra is computed exactly. As an
application of our results the certain combinatorial identity is
derived.