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From Aztec diamonds to pyramids: steep tilings
We introduce a family of domino tilings that includes tilings of the Aztec
diamond and pyramid partitions as special cases. These tilings live in a strip
of of the form for some integer , and are parametrized by a binary word that
encodes some periodicity conditions at infinity. Aztec diamond and pyramid
partitions correspond respectively to and to the limit case
. For each word and for different types of boundary
conditions, we obtain a nice product formula for the generating function of the
associated tilings with respect to the number of flips, that admits a natural
multivariate generalization. The main tools are a bijective correspondence with
sequences of interlaced partitions and the vertex operator formalism (which we
slightly extend in order to handle Littlewood-type identities). In
probabilistic terms our tilings map to Schur processes of different types
(standard, Pfaffian and periodic). We also introduce a more general model that
interpolates between domino tilings and plane partitions.Comment: 36 pages, 22 figures (v3: final accepted version with new Figure 6,
new improved proof of Proposition 11
Littlewood--Richardson coefficients and integrable tilings
We provide direct proofs of product and coproduct formulae for Schur
functions where the coefficients (Littlewood--Richardson coefficients) are
defined as counting puzzles. The product formula includes a second alphabet for
the Schur functions, allowing in particular to recover formulae of
[Molev--Sagan '99] and [Knutson--Tao '03] for factorial Schur functions. The
method is based on the quantum integrability of the underlying tiling model.Comment: 29 pages, color figures. v2: corrected minor misprints and added
short appendix. v3: fig 16 fixe
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