Permutations and Combinations with inner products. : Need help!

Question

I have two strings

String A : $\langle e_{1}|\langle e_{2}|\langle e_{2}| \langle e_{2}|\langle e_{2}|\langle e_{2}| $

String B : $|f_{1}\rangle |f_{1}\rangle |f_{1}\rangle |f_{1}\rangle |f_{2}\rangle |f_{2}\rangle $

String B is combined with different permutations of string A. The permutations of string A are -

1. $\langle e_{1}|\langle e_{2}|\langle e_{2}| \langle e_{2}|\langle e_{2}|\langle e_{2}| $

2. $\langle e_{2}|\langle e_{1}|\langle e_{2}| \langle e_{2}|\langle e_{2}|\langle e_{2}| $

3. $\langle e_{2}|\langle e_{2}|\langle e_{1}| \langle e_{2}|\langle e_{2}|\langle e_{2}| $

4. $\langle e_{2}|\langle e_{2}|\langle e_{2}| \langle e_{1}|\langle e_{2}|\langle e_{2}| $

5. $\langle e_{2}|\langle e_{2}|\langle e_{2}| \langle e_{2}|\langle e_{1}|\langle e_{2}| $

6. $\langle e_{2}|\langle e_{2}|\langle e_{2}| \langle e_{2}|\langle e_{2}|\langle e_{1}| $

String B is combined with each of the permutations from 1 to 6.

In this way for example :

$\langle e_{1}|\langle e_{2}|\langle e_{2}| \langle e_{2}|\langle e_{2}|\langle e_{2}| $ $|f_{1}\rangle |f_{1}\rangle |f_{1}\rangle |f_{1}\rangle |f_{2}\rangle |f_{2}\rangle $ = ${\langle e_{2}|f_{1}\rangle}^4 {\langle e_{2}|f_{2}\rangle}^1 {\langle e_{1}|f_{2}\rangle}^1$ and

$\langle e_{2}|\langle e_{2}|\langle e_{1}| \langle e_{2}|\langle e_{2}|\langle e_{2}| $ $|f_{1}\rangle |f_{1}\rangle |f_{1}\rangle |f_{1}\rangle |f_{2}\rangle |f_{2}\rangle $$={\langle e_{2}|f_{1}\rangle}^3 {\langle e_{2}|f_{2}\rangle}^2 {\langle e_{1}|f_{1}\rangle}^1$

Basically, the combination happens between the end of string A and the beginning of the B string, and it goes on, outward

And you get two different kinds of strings after A and B combine -

String of type 1 : ${\langle e_{2}|f_{1}\rangle}^3 {\langle e_{2}|f_{2}\rangle}^2 {\langle e_{1}|f_{1}\rangle}^1$ which occurs 4 times.

And string of type 2 : ${\langle e_{2}|f_{1}\rangle}^4 {\langle e_{2}|f_{2}\rangle}^1 {\langle e_{1}|f_{2}\rangle}^1$ which occurs twice.

My question is this. Without going through the calculations I can say, there are two types of strings which occur. ${\langle e_{2}|f_{1}\rangle}^3 {\langle e_{2}|f_{2}\rangle}^2 {\langle e_{1}|f_{1}\rangle}^1$ and ${\langle e_{2}|f_{1}\rangle}^4 {\langle e_{2}|f_{2}\rangle}^1 {\langle e_{1}|f_{2}\rangle}^1$ But I cannot predict that the string of type 1 occurs 4 times and string of type 2 occurs twice. How can I predict this? How can we do this using combinatorics?

Answer 1

Let non-negative integers $n_{11},n_{12},n_{21},n_{22}$ be admissible counts of ${\langle e_{1}|f_{1}\rangle},{\langle e_{1}|f_{2}\rangle},{\langle e_{2}|f_{1}\rangle},{\langle e_{2}|f_{2}\rangle}$, respectively. The counts are admissible if and only if $$ n_{11}+n_{12}=\#e_1,\;n_{21}+n_{22}=\#e_2,\;n_{11}+n_{21}=\#f_1,\;n_{12}+n_{22}=\#f_2. $$

Then the number of sequences with the given admissible counts is: $$ \binom{n_{11}+n_{21}}{n_{11}}\binom{n_{12}+n_{22}}{n_{22}}, $$ which corresponds to the number of ways to choose the positions of the corresponding pairs.