Séminaire Lotharingien de Combinatoire, 91B.2 (2024), 12 pp.
Colin Defant, Rachana Madhukara and Hugh Thomas
Toric and Permutoric Promotion
Abstract.
We introduce toric promotion as a cyclic analogue of
Schützenberger's promotion operator. Toric promotion acts on the set
of labelings of a graph G; it is defined as the composition of
certain toggle operators, listed in a natural cyclic order. We provide
a surprisingly simple description of the orbit structure of toric
promotion when G is a forest. We then consider more general
permutoric promotion operators, which are defined as compositions of
the same toggle operators, but in permuted orders. When G is a path
graph, we provide a complete description of the orbit structures of
all permutoric promotion operators, showing that they satisfy the
cyclic sieving phenomenon.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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