Abstract
We initiate the study of the enumerative combinatorics of the intervals in the Dyck pattern poset. More specifically, we find some closed formulas to express the size of some specific intervals, as well as the number of their covering relations. In most of the cases, we are also able to refine our formulas by rank. We also provide the first results on the Möbius function of the Dyck pattern poset, giving for instance a closed expression for the Möbius function of initial intervals whose maximum is a Dyck path having exactly two peaks.
| Original language | English |
|---|---|
| Pages (from-to) | 473-487 |
| Number of pages | 15 |
| Journal | Order |
| Volume | 38 |
| Issue number | 3 |
| Early online date | 3 Apr 2021 |
| DOIs | |
| Publication status | Published - 31 Oct 2021 |
Funding
A.B. and L.F. are members of the INdAM Research group GNCS; they are partially supported by INdAM -GNCS 2019 project “Studio di proprietá combinatoriche di linguaggi formali ispirate dalla biologia e da strutture bidimensionali” and by a grant of the “Fondazione della Cassa di Risparmio di Firenze” for the project “Rilevamento di pattern: applicazioni a memorizzazione basata sul DNA, evoluzione del genoma, scelta sociale” E. S. is partially supported by a Leverhulme Research Fellowship.
Keywords
- Dyck path
- enumeration
- interval
- Möbius function
- poset