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Fix equation rendering #38

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Merged
FanwangM merged 3 commits into from
Apr 23, 2026
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Fix equation rendering #38

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f5f6974
Fix equation rendering
FanwangM Apr 23, 2026
c317963
Use https instead of http
FanwangM Apr 23, 2026
c719bb6
Fix the misaligned equations
FanwangM Apr 23, 2026
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27 changes: 15 additions & 12 deletions README.md
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Original file line number Diff line number Diff line change
@@ -1,4 +1,4 @@
[![Python 3](http://img.shields.io/badge/python-3-blue.svg)](https://docs.python.org/3/)
[![This project supports Python 3.10+](https://img.shields.io/badge/Python-3.10+-blue.svg)](https://python.org/downloads)
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Copilot AI Apr 23, 2026

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The Shields.io badge URL includes a literal "+" in the path ("3.10+"). Shields generally expects special characters to be URL-encoded; otherwise the badge can fail to render in some contexts. Consider encoding it as "%2B" ("3.10%2B") to make the badge URL robust.

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[![pre-commit](https://img.shields.io/badge/pre--commit-enabled-brightgreen?logo=pre-commit&logoColor=white)](https://github.com/theochem/matrix-permanent/actions/workflows/pull_request.yml)
[![GNU GPLv3](https://img.shields.io/badge/license-%20%20GNU%20GPLv3%20-green?style=plastic)](https://www.gnu.org/licenses/gpl-3.0.en.html)

Expand Down Expand Up @@ -37,9 +37,10 @@ Compute the permanent of a matrix using the best algorithm for the shape of the
Compute the permanent of a matrix combinatorically.

**Formula:**
```math

$$
\text{per}(A) = \sum_{\sigma \in P(N,M)}{\prod_{i=1}^M{a_{i,{\sigma(i)}}}}
```
$$

**Parameters:**

Expand All @@ -55,29 +56,31 @@ Compute the permanent of a matrix combinatorically.

**Formula:**

```math
$$
\text{per}(A) = \frac{1}{2^{N-1}} \cdot \sum_{\delta \in \left[\delta_1 = 1,~ \delta_2 \dots \delta_N=\pm1\right]}{
\left(\sum_{k=1}^N{\delta_k}\right){\prod_{j=1}^N{\sum_{i=1}^N{\delta_i a_{i,j}}}}}
Comment on lines 60 to 61
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Copilot AI Apr 23, 2026

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These equations are laid out across multiple lines in Markdown, but KaTeX/GitHub math rendering treats newlines inside $$ ... $$ as whitespace (not line breaks), so this will likely render as a single very long line and overflow on GitHub. Consider using an aligned/multline environment with explicit \\ line breaks to preserve the intended formatting.

Suggested change
\text{per}(A) = \frac{1}{2^{N-1}} \cdot \sum_{\delta \in \left[\delta_1 = 1,~ \delta_2 \dots \delta_N=\pm1\right]}{
\left(\sum_{k=1}^N{\delta_k}\right){\prod_{j=1}^N{\sum_{i=1}^N{\delta_i a_{i,j}}}}}
\begin{aligned}
\text{per}(A) &= \frac{1}{2^{N-1}} \cdot \sum_{\delta \in \left[\delta_1 = 1,~ \delta_2 \dots \delta_N=\pm1\right]}{ \\
&\quad \left(\sum_{k=1}^N{\delta_k}\right){\prod_{j=1}^N{\sum_{i=1}^N{\delta_i a_{i,j}}}}}
\end{aligned}

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```
$$

**Additional Information:**
The original formula has been generalized here to work with $M$-by-$N$ rectangular permanents with
$M \leq N$ by use of the following identity (shown here for $M \geq N$):

```math
$$
\begin{aligned}
\text{per}\left(\begin{matrix}a_{1,1} & \cdots & a_{1,N} \\ \vdots & \ddots & \vdots \\ a_{M,1} & \cdots & a_{M,N}\end{matrix}\right) = \frac{1}{(M - N + 1)!} \cdot \text{per}\left(\begin{matrix}a_{1,1} & \cdots & a_{1,N} & 1_{1,N+1} & \cdots & 1_{1,M} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ a_{M,1} & \cdots & a_{M,N} & 1_{M,N+1} & \cdots & 1_{M,M}\end{matrix}\right)
```
\end{aligned}
$$
Comment on lines +68 to +72
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Copilot AI Apr 23, 2026

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This matrix identity is extremely wide; with $$ ... $$ and no explicit line breaks it will likely overflow horizontally in GitHub’s renderer. Consider splitting it across multiple lines using aligned/multline (and \\) so it remains readable on narrow screens.

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This can be neatly fit into the original formula by extending the inner sums over $\delta$ from $[1,M]$ to $[1,N]$:

```math
$$
\text{per}(A) = \frac{1}{2^{N-1}} \cdot \frac{1}{(N - M + 1)!}\cdot \sum_{\delta \in \left[\delta_1 = 1,~ \delta_2 \dots \delta_N=\pm1\right]}{
\left(\sum_{k=1}^N{\delta_k}\right)
\prod_{j=1}^N{\left(
\sum_{i=1}^M{\delta_i a_{i,j}} + \sum_{i=M+1}^N{\delta_i}
\right)}
}
```
$$

**Parameters:**

Expand All @@ -93,7 +96,7 @@ This can be neatly fit into the original formula by extending the inner sums ove

**Formula:**

```math
$$
\text{per}(A) = \sum_{k=0}^{M-1}{
{(-1)}^k
\binom{N - M + k}{k}
Expand All @@ -103,7 +106,7 @@ This can be neatly fit into the original formula by extending the inner sums ove
}
}
}
```
$$
Comment on lines 99 to +109
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Copilot AI Apr 23, 2026

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This multi-line formula will render as a single line under KaTeX unless you add an aligned/multline environment and explicit \\ breaks. As-is, it’s likely to overflow horizontally and be hard to read on GitHub.

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**Parameters:**

Expand Down Expand Up @@ -181,4 +184,4 @@ is with pip.
# License

This code is distributed under the GNU General Public License version 3 (GPLv3).
See <http://www.gnu.org/licenses/> for more information.
See <https://www.gnu.org/licenses/> for more information.
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