A quick proof that $321$-avoiding permutations without double deficiencies are counted by the Motzkin numbers

Tipaluck Krityakierne; Thotsaporn "Aek" Thanatipanonda; Doron Zeilberger

A quick proof that $321$-avoiding permutations without double deficiencies are counted by the Motzkin numbers

Combinatorics 2026-07-03 v1

Authors: Tipaluck Krityakierne , Thotsaporn "Aek" Thanatipanonda , Doron Zeilberger

Abstract

We present a short "proof from the Book" that $321$ -avoiding permutations without double deficiencies are counted by the Motzkin numbers. Although this result was first proved by Rubey and Stump, our proof gives a much simpler and more transparent description of the same bijection.

Keywords

321-avoiding permutations double deficiencies Motzkin numbers bijection combinatorial proof

Cite

@article{arxiv.2607.05431,
  title  = {A quick proof that $321$-avoiding permutations without double deficiencies are counted by the Motzkin numbers},
  author = {Tipaluck Krityakierne and Thotsaporn "Aek" Thanatipanonda and Doron Zeilberger},
  journal= {arXiv preprint arXiv:2607.05431},
  year   = {2026}
}