A finite unit interval order, also called a semiorder, is an interval order that can be represented by closed intervals
all of length 1 (Luce 1956, Scott and Suppes 1958). For a representation with intervals
![[a_i,a_i+1]](/images/equations/UnitIntervalOrder/Inline1.svg)
indexed by nondecreasing left endpoints, the strict order is 
precisely when 
. Joining each incomparable
pair gives a natural unit interval graph
on these ordered vertices.
A unit interval order 
on 
elements has two Dyck path encodings
(Gélinas et al. 2026). In the first path 
, the unit squares between
the path and the diagonal are exactly ![[j-1,j]×[i-1,i]](/images/equations/UnitIntervalOrder/Inline7.svg)
for incomparable 
, using the indexing by left endpoints. For the
second path, let 
count the unit squares between the path and the diagonal
in column 
from the left. The word 
defines a partially ordered
set by 
precisely when 
or when 
and 
. The path 
is the unique Dyck path whose
word defines an order isomorphic to 
.
Haglund's zeta map 
on a Dyck path 
below the diagonal is obtained by reading the terminal points
of its steps on successive lines 
for 
, 1, ..., from southwest to
northeast on each line, and exchanging horizontal and vertical steps in the resulting
list (Haglund 2008, Gélinas et al. 2026). Matherne et al. (2024)
conjectured that
 |
Gélinas et al. (2026) give an inductive proof, and Fang (2024) gives a bijective proof.
-Catalan
Numbers and the Space of Diagonal Harmonics. Providence, RI: Amer. Math. Soc.,
2008.