The Pop-stack-sorting Operator on Tamari Lattices

Publication date: 24 January 2022

Abstract: Motivated by the pop-stack-sorting map on the symmetric groups, Defant defined an operator

m a t h s f {P o p}_{M} : M o M

for each complete meet-semilattice

M

by

mathsf{Pop}_M(x)=�igwedge({yin M: ylessdot x}cup {x}).

This paper concerns the dynamics of

m a t h s f {P o p}_{m a t h r m {T a m}_{n}}

, where

m a t h r m {T a m}_{n}

is the

n

-th Tamari lattice. We say an element

x i n m a t h r m {T a m}_{n}

is

t

-

m a t h s f P o p

-sortable if

m a t h s f {P o p}_{M}^{t} (x)

is the minimal element and we let

h_{t} (n)

denote the number of

t

-

m a t h s f P o p

-sortable elements in

m a t h r m {T a m}_{n}

. We find an explicit formula for the generating function

s u m_{n g e 1} h_{t} (n) z^{n}

and verify Defant's conjecture that it is rational. We furthermore prove that the size of the image of

m a t h s f {P o p}_{m a t h r m {T a m}_{n}}

is the Motzkin number

M_{n}

, settling a conjecture of Defant and Williams.

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