dhondt-formula.tex

\documentclass{standalone}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{cancel}
\begin{document}
$\begin{array}{l}
\displaystyle
\frac{
\displaystyle
\sum_{\substack{(m_1,\ldots,m_r)\in\mathbb{N}^r\\m_1+\cdots+m_r=N}}
\left(
m_r\cdot
\left.\sum_{j=1}^r m_j \int_{\prod_{i\neq j}[m_i,m_i+1]}
\frac{x_r^{k-1}}{(x_1+\cdots+x_r)^n}
\right|_{x_j=m_j}
\,dx_1\cdots\cancel{dx_j}\cdots dx_r
\right)
}{
\displaystyle
\sum_{\substack{(m_1,\ldots,m_r)\in\mathbb{N}^r\\m_1+\cdots+m_r=N}}
\left(
\left.\sum_{j=1}^r m_j \int_{\prod_{i\neq j}[m_i,m_i+1]}
\frac{x_r^{k-1}}{(x_1+\cdots+x_r)^n}
\right|_{x_j=m_j}
\,dx_1\cdots\cancel{dx_j}\cdots dx_r
\right)
}
-
\frac{Nk}{n}
\\\strut\\
\text{with~}r=n-k+1
\end{array}
$
\end{document}

dhondt-integrals.m

n=5;k=4;r=n-k+1
I1 = Simplify[m1 Integrate[x2^(k-1)/(m1+x2)^n, {x2,m2,m2+1}]]
I2 = Simplify[m2 Integrate[m2^(k-1)/(x1+m2)^n, {x1,m1,m1+1}]]
Itot = Simplify[I1+I2]
Simplify[Itot /. m2->(bigN-m1)]
ev0 = Simplify[Sum[%,{m1,0,bigN}]]
Simplify[m2 Itot /. m2->(bigN-m1)]
ev1 = Simplify[Sum[%,{m1,0,bigN}]]
Simplify[ev1/ev0 - k bigN/n]
% == bigN(9bigN^2+17bigN+7)/(30(bigN+1)^3)


n=5;k=3;r=n-k+1
I1 = Simplify[m1 Integrate[x3^(k-1)/(m1+x2+x3)^n, {x2,m2,m2+1}, {x3,m3,m3+1}]]
I2 = Simplify[m2 Integrate[x3^(k-1)/(x1+m2+x3)^n, {x1,m1,m1+1}, {x3,m3,m3+1}]]
I3 = Simplify[m3 Integrate[m3^(k-1)/(x1+x2+m3)^n, {x1,m1,m1+1}, {x2,m2,m2+1}]]
Itot = Simplify[I1+I2+I3]
Simplify[Itot /. m3->(bigN-m1-m2)]
Simplify[Sum[%,{m2,0,bigN-m1}]]
ev0 = Simplify[Sum[%,{m1,0,bigN}]]
Simplify[m3 Itot /. m3->(bigN-m1-m2)]
Simplify[Sum[%,{m2,0,bigN-m1}]]
ev1 = Simplify[Sum[%,{m1,0,bigN}]]
Simplify[ev1/ev0 - k bigN/n]
% == bigN(24bigN^3+114bigN^2+177bigN+89)/(60(bigN+1)^2(bigN+2)^2)


n=5;k=2;r=n-k+1
I1 = Simplify[m1 Integrate[x4^(k-1)/(m1+x2+x3+x4)^n, {x2,m2,m2+1}, {x3,m3,m3+1}, {x4,m4,m4+1}]]
I2 = Simplify[m2 Integrate[x4^(k-1)/(x1+m2+x3+x4)^n, {x1,m1,m1+1}, {x3,m3,m3+1}, {x4,m4,m4+1}]]
I3 = Simplify[m3 Integrate[x4^(k-1)/(x1+x2+m3+x4)^n, {x1,m1,m1+1}, {x2,m2,m2+1}, {x4,m4,m4+1}]]
I4 = Simplify[m4 Integrate[m4^(k-1)/(x1+x2+x3+m4)^n, {x1,m1,m1+1}, {x2,m2,m2+1}, {x3,m3,m3+1}]]
Itot = Simplify[I1+I2+I3+I4]
Simplify[Itot /. m4->(bigN-m1-m2-m3)]
Simplify[Sum[%,{m3,0,bigN-m2-m1}]]
Simplify[Sum[%,{m2,0,bigN-m1}]]
ev0 = Simplify[Sum[%,{m1,0,bigN}]]
Simplify[m4 Itot /. m4->(bigN-m1-m2-m3)]
Simplify[Sum[%,{m3,0,bigN-m2-m1}]]
Simplify[Sum[%,{m2,0,bigN-m1}]]
ev1 = Simplify[Sum[%,{m1,0,bigN}]]
Simplify[ev1/ev0 - k bigN/n]
% == bigN(3bigN^2+13bigN+13)/(10(bigN+1)(bigN+2)(bigN+3))

See https://twitter.com/gro_tsen/status/1656632281325346816 and https://mathoverflow.net/questions/446507/grouping-lists-together-in-a-proportional-election-image-of-a-dirichlet-distrib for context