Table 3.

Dynamic programming matrix with distributions of the number of adjacent UMIs

$\frac{S_g}{K}$	1	2	3	…	S g
0	1	1 − pNeighb	(1 − pNeighb)2	...	${\left(1-p_{\mathit{\text{Neighb}}}\right)}^{S_g-1}$
1	0	p Neighb	$\left(1-p_{\mathit{\text{Neighb}}}\right)\ast p_{\mathit{\text{Neighb}}}++p_{\mathit{\text{Neighb}}}\ast \left(1-p_{\mathit{\text{Neighb}}}\frac{K-1}{K}\right)$	…	$t_{0,S-1}\ast p_{\mathit{\text{Neighb}}}++t_{1,S-1}\ast \left(1-p_{\mathit{\text{Neighb}}}\frac{K-1}{K}\right)$
2	0	0	$p_{\mathit{\text{Neighb}}}^2\frac{K-1}{K}$	…	$t_{1,S-1}\ast p_{\mathit{\text{Neighb}}}\frac{K-1}{K}++t_{2,S-1}\ast \left(1-p_{\mathit{\text{Neighb}}}\frac{K-2}{K}\right)$
…	…	…	…	…	…
k	0	0	0	…	$t_{k-1,S-1}\ast p_{\mathit{\text{Neighb}}}\frac{K-k+1}{K}++t_{k,S-1}\ast \left(1-p_{\mathit{\text{Neighb}}}\frac{K-k}{K}\right)$
…	…	…	…	…	…
K	0	0	0	…	$t_{K-1,S-1}\frac{p_{\mathit{\text{Neighb}}}}{K}+t_{K,S-1}$

Here, K is the maximum number of adjacent UMIs, S_g is the maximum number of molecules per gene. A cell t_{k, s} of the matrix contains probability of observing k adjacent UMIs for a fixed UMI in a cell of size s