I'm trying to understand the following things.
Let $\{1, \dots , n\}:=[n]$ and consider multi-indices $\textbf{a,b} \in [n]^l$ for some integer $l$.
Let's define an equivalence relation $\textbf{a}\sim \textbf{b} \Leftrightarrow \textbf{a}_i = \textbf{a}_j \Leftrightarrow \textbf{b}_i = \textbf{b}_j \, \, \forall i,j \in [l].$ (equality patterns)
Which are the equivalence classes of a such relation? Can you provide me some simple examples even with $l=2$?
Let's say that $n=2, l=2$.
so if $\textbf{a} = (1,1) \Rightarrow \textbf{b} = (1,1)$ or $\textbf{b} = (2,2)$
if $\textbf{a} = (2,2) \Rightarrow \textbf{b} = (1,1)$ or $\textbf{b} = (2,2)$
if $\textbf{a} = (1,2) \Rightarrow \textbf{b} = (1,2)$ or $\textbf{b} = (2,1)$
if $\textbf{a} = (2,1) \Rightarrow \textbf{b} = (1,2)$ or $\textbf{b} = (2,1)$
This is what I manage to get but I still have several perplexities about which the equivalence classes are in this case.
To summarize the discussion in the comments:
Given an $l-$tuple, $(a_1, \cdots, a_l)$ we define a partition of $\{1, \cdots l\}$ according to the digits in the tuple.
Thus, with $l=7, n=5$, the $6-$tuple $(3,3,5,1,3,5,3)$ we would get the partition $\{(1,2,5,7),\,(3,6), (4)\}$.
Then: two $l-$tuples are equivalent iff they define the same partition.
Indeed, the partition tells you precisely which "blocks" of indices must have a common digit.
Note: as an example, take $l=2$. There are only two partitions of $\{1,2\}$, namely $\{(1,2)\}$ and $\{(1),(2)\}$. The first gives you the pairs of the form $(a,a)$ and the second gives you the pairs of the form $(a,b)$ with $a\neq b$.