27 research outputs found
Non-commutative Sylvester's determinantal identity
Sylvester's identity is a classical determinantal identity with a
straightforward linear algebra proof. We present a new, combinatorial proof of
the identity, prove several non-commutative versions, and find a
-extension that is both a generalization of Sylvester's identity and the
-extension of the MacMahon master theorem.Comment: 28 pages, 8 figure
The weighted hook-length formula II: Complementary formulas
Recently, a new weighted generalization of the branching rule for the hook
lengths, equivalent to the hook formula, was proved. In this paper, we
generalize the complementary branching rule, which can be used to prove
Burnside's formula. We present three different proofs: bijective, via weighted
hook walks, and via the ordinary weighted branching rule.Comment: 20 pages, 9 figure
Non-commutative extensions of the MacMahon Master Theorem
We present several non-commutative extensions of the MacMahon Master Theorem,
further extending the results of Cartier-Foata and Garoufalidis-Le-Zeilberger.
The proofs are combinatorial and new even in the classical cases. We also give
applications to the -extension and Krattenthaler-Schlosser's
-analogue.Comment: 28 pages, 6 figure
Geometry and complexity of O'Hara's algorithm
In this paper we analyze O'Hara's partition bijection. We present three type
of results. First, we show that O'Hara's bijection can be viewed geometrically
as a certain scissor congruence type result. Second, we obtain a number of new
complexity bounds, proving that O'Hara's bijection is efficient in several
special cases and mildly exponential in general. Finally, we prove that for
identities with finite support, the map of the O'Hara's bijection can be
computed in polynomial time, i.e. much more efficiently than by O'Hara's
construction.Comment: 20 pages, 4 figure