calculate the cardinality of $B_{k,\ell}$. Suppose $((i_1, i_2,\ldots,i_{k}), (j_1,j_2,\ldots,j_{\ell}))\in B_{k,\ell}$ with $|\{i_1,\ldots,i_{k}\}\cap \{j_1,\ldots,j_{\ell}\}|=m$, for some $m=1,2,\ldots, \min\{k,\ell\}$, where $|\{\cdot\}|$ denotes cardinality of the set $\{\cdot\}$. Therefore typical element of $B_{k,\ell}$ will look like $$((d_1, d_2,\ldots, d_m, i_{m+1}, \ldots, i_{k}), (d_1,d_2,\ldots, d_m, j_{m+1}, \ldots, j_{\ell})).$$ Observe that, we shall get maximum number of free entries in $B_{k,\ell}$, if following conditions hold

1.  each elements of $ \{d_1, d_2, \ldots, d_{m}\}$ are distinct,

2.  if $k-m$ is even. Then $(i_{m+1}, \ldots, i_{k})$ is [*opposite sign pair matched*]{} with $ \{d_1, d_2, \ldots, d_{m}\} \cap \{i_{m+1},\ldots,i_{k}\}=\emptyset$. Similar condition also hold when $\ell-m$ is even,

3.  if $k-m$ is odd. Then $\{i_{m+1},\ldots,i_{k}\} \setminus \{i^*\}$ is [*opposite sign pair matched*]{} and $ \{d_1, d_2, \ldots, d_{m-2}\} \cap \{i_{m+1},\ldots,i_{k}\} \setminus \{i^*\}=\emptyset$, where $i^{*}$ is [*opposite sign pair matched*]{} with $d_s$ for some $s=1,2, \ldots, m$. Similar condition also hold when $\ell-m$ is odd.

Under the above assumption, the cardinality of $B_{k,\ell}$ will be

$$\begin{aligned}
\label{card_B_k_l}
     |B_{k,\ell}|  & = \left\{\begin{array}{ll}      
             O(n^{m-1+\frac{k-m}{2} +\frac{\ell -m}{2}}) & \text{if}\ (k-m) \mbox{ and } (\ell-m) \mbox{ both are even,}\\\\
              O(n^{m-3+\frac{k-m+1}{2} +\frac{\ell -m+1}{2}}) & \text{if}\ (k-m) \mbox{ and } (\ell-m) \mbox{ both are odd,}\\\\
            O(n^{m-2+\frac{k-m+\ell -m+1}{2}})& \text{otherwise},        
              \end{array}\right.     \nonumber \\
     & = \left\{\begin{array}{ll}    
             