I have the next problem: "There are 11 workers and a fixed week such that 1) Every day, three workers rest and 8 workers work. 2) Each worker rests exactly two days. And
3a. Each worker can rest in consecutive days.
3b. Not 3a.
In how many ways we can distribute the work schedule?"
Now, let $a_{i,j}\in\{0,1\}$ defined by $a_{i,j}=1$ if worker $j$ works at day $i$ and 0 in other case. In 3a case, a schedule is posible if and only if there is a matrix $A$ such that $$A=\left(\begin{array}{ccccc}a_{1,1}&a_{1,2}&a_{1,3}&...&a_{1,11}\\a_{2,1}&a_{2,2}&a_{2,3}&...&a_{2,11}\\\vdots&\vdots&\vdots&\vdots&\vdots\\a_{7,1}&a_{7,2}&a_{7,3}&...&a_{7,11}\\\end{array}\right)$$ where, for fixed $i$, we have $\sum_{j=1}^{11}a_{i,j}=8$ and, for fixed $j$ we have $\sum_{i=1}^{7}a_{i,j}=2$.
Now, let $X$ all matrices of the above form, and $\mathbb{T}$ the set of all matrix operations defined on the set of matrices with $7$ columns and $11$ rows such that $T(M)$ is matrix $M$ but with two interchanged rows or two interchanged columns.
It is clear that if $A\in X$ then $T(A)\in X$ for every $T\in\mathbb{T}$. Now, I want to define a group $G$ generated in some sense by $\mathbb{T}$ such that $\# X=\sum_{T\in G}|G:\mathrm{St}(T)|$.
Any hint will be apreciated.