# Counting majorizations

Let $\mathbf{x} = (x_1,\dots, x_n)$ and $\mathbf{y} = (y_1, \dots, y_n)$ be two increasing positive integer sequences. We say that $\mathbf{y}$ majorizes $\mathbf{x}$ if $$\sum_{i=1}^k x_i \leq \sum_{i=1}^k y_i$$ for all $k=1,\dots, n$ with equality if $k=n$.

If I fix an increasing positive integer sequence $\mathbf{y} = (y_1, \dots, y_n)$, then how many increasing positive integer sequences $\mathbf{x} = (x_1, \dots, x_n)$ are majorized by $\mathbf{y}$?

• Unless you're redefining things, majorization requires all $x_i, y_i$ to be also sorted in non-descending orde before you use the inequality conditions. – Macavity Jul 4 '17 at 2:14
• Thanks, good point. – user353673 Jul 4 '17 at 2:19