10,806 research outputs found
An extension of Tamari lattices
For any finite path on the square grid consisting of north and east unit
steps, starting at (0,0), we construct a poset Tam that consists of all
the paths weakly above with the same number of north and east steps as .
For particular choices of , we recover the traditional Tamari lattice and
the -Tamari lattice.
Let be the path obtained from by reading the unit
steps of in reverse order, replacing the east steps by north steps and vice
versa. We show that the poset Tam is isomorphic to the dual of the poset
Tam. We do so by showing bijectively that the poset
Tam is isomorphic to the poset based on rotation of full binary trees with
the fixed canopy , from which the duality follows easily. This also shows
that Tam is a lattice for any path . We also obtain as a corollary of
this bijection that the usual Tamari lattice, based on Dyck paths of height
, is a partition of the (smaller) lattices Tam, where the are all
the paths on the square grid that consist of unit steps.
We explain possible connections between the poset Tam and (the
combinatorics of) the generalized diagonal coinvariant spaces of the symmetric
group.Comment: 18 page
Free cumulants, Schr\"oder trees, and operads
The functional equation defining the free cumulants in free probability is
lifted successively to the noncommutative Fa\`a di Bruno algebra, and then to
the group of a free operad over Schr\"oder trees. This leads to new
combinatorial expressions, which remain valid for operator-valued free
probability. Specializations of these expressions give back Speicher's formula
in terms of noncrossing partitions, and its interpretation in terms of
characters due to Ebrahimi-Fard and Patras.Comment: 23 page
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