Let $n,\ell \geq 2$ be integers. For an integer $k$ with $1 \leq k \leq n$ and two multisets $A, B$ each consisting of $\ell$ numbers in $[n] = \{1, \ldots, n\}$, we say that $A, B$ satisfy the pattern $R_k$, and write $A R_k B$, if the following two conditions hold:
(1) The elements of $A$ are the same as those of $B$, except for exactly two.
(2) Let $a_1, a_2, b_1, b_2$ be the two elements in $A,B$, respectively, which differ. Then $(b_1, b_2) = (a_1 - k, a_2 + k+1)$ or $(b_2, b_1) = (a_1 - k, a_2 + k+1)$.
For example, let $A = \{\{ 4,1,1,1,2,2,3,3,3\}\}$ and $B = \{\{3,3,1,1,2,2,3,3,3\}\}$. Then $AR_1 B$ with $a_1 = 4$, $a_2 = 1$, $b_1 = b_2 = 3$.
Now for a given number $m$, let $A_1, \ldots, A_m$ be given multisets, each of size $\ell$ consisting of numbers in $[n]$. I am interested in obtaining an upper bound for $$ S = \#\{(k, i,j) \in [n] \times [m]^2 \ : \ A_i R_k A_j \} $$ in terms of $m,n,\ell$.
Perhaps a probabilistic method works here? Maybe starting with this: Choose $m$ multisets $A_1, \ldots, A_m$ from $[n]$ uniformly at random, each of size $\ell$. Let $$ X_{i,j,k} = \begin{cases} 1 & \mbox{ if } A_i R_k A_j\\ 0 & \mbox{ otherwise.} \end{cases} $$ Now let $$X = \sum_{k=1}^n \sum_{1 \leq i,j \leq m} X_{i,j,k}.$$ Now $U = U(m,n,\ell)$ is an upper bound for $S$ if and only if $\mathbb{P}(X \leq U) = 1$. Perhaps the Chebyshev's inequality could work here? Note we need to find a $U$ such that $\mathbb{P}(X \leq U) \geq 1$.