From Aztec diamonds to pyramids: steep tilings

TL;DR

This paper introduces a new family of domino tilings that generalize Aztec diamonds and pyramid partitions, providing explicit generating functions and connecting to Schur processes and interlaced partitions.

Contribution

It defines a unified framework for domino tilings parametrized by binary words, extending known models and deriving product formulas for their generating functions.

Findings

Product formulas for tiling generating functions

Connection to Schur processes and interlaced partitions

Interpolation between domino tilings and plane partitions

Abstract

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 $\mathbb{Z}^2$ of the form $1 \leq x-y \leq 2\ell$ for some integer $\ell \geq 1$, and are parametrized by a binary word $w\in\{+,-\}^{2\ell}$ that encodes some periodicity conditions at infinity. Aztec diamond and pyramid partitions correspond respectively to $w=(+-)^\ell$ and to the limit case $w=+^\infty-^\infty$. For each word $w$ 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.