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Séminaire Lotharingien de Combinatoire, 78B.60 (2017), 12 pp.

# Florian Aigner

# Refined Enumerations of Alternating Sign Triangles

**Abstract.**
We generalise alternating sign triangles, which were recently
introduced by Ayyer, Behrend and Fischer and shown to be equinumerous
with ASMs, to AS-trapezoids and introduce a refinement by Catalan
objects. We show that the number of AS-trapezoids associated with a
fixed Catalan object is a polynomial in the length of the shorter base
of the trapezoid. As a special case we obtain an analogue of a
polynomiality theorem for fully packed loops with nested arches.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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