Obtaining a poset of matrices in Sage

asked 2021-06-03 23:55:27 +0100

klaaa gravatar image

Let $R=(r_1,...,r_m)$ and $S=(S_1,...,s_n)$ be a sequence of nonnegative integers with $r_1+...+r_m=s_1+...+s_n$. Let $N(R,S)$ denote the set of all $m \times n$-matrices with nonnegative integer entries whose row sums are given by R and whose column sums are given by S.

For an mxn matrix $A=(a_{i,j})$ define the mxn matrix $\Sigma(A)=(\sigma_{i,j})$ with entries $\sigma_{i,j}= \sum\limits_{k=1}^{i}{\sum\limits_{l=1}^{j}{a_{k,l}}}$.

Define an order on $N(R,S)$ by saying that $A_1 \leq A_2$ if and only if $\Sigma(A_1) \geq \Sigma(A_2)$, where the last $\geq $ means the entrywise comparison.

For example for $R=S=(1,1,...,1)$ we obtain the poset of the strong Bruhat order for the symmetric group.

Question: Is there an easy way to obtain the poset $N(R,S)$ in Sage?

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