Math is often about finding the right analogy, often a spatial analogy. \u00a0Suppose you want to compute the sum of integers, $latex S = 1+2+3+ \\hdots + N$. Consider decomposing the sum as,<\/p>\n
$latex \\begin{array}{cccccc}S = & 1 & + 1 & + 1 & + \\hdots & + 1\\\\ & & + 1 & + 1 & + \\hdots & + 1 \\\\ & & & + 1 & + \\hdots & + 1\\\\ & & & & \\vdots & \\\\ & & & & & + 1,\\end{array}$<\/p>\n
where you’ll note the column sums equal the integers in the sequence. \u00a0You’ll have a bunch of ones that form a triangle. Imagine taking such triangle, copying it, flipping it and then joining the copy to the first triangle. Removing the “$latex +$” signs you’d get something that looks like,<\/p>\n
$latex \\begin{array}{ccccc} 1 & 1 & 1 & \\hdots & 1\\\\ (1) & 1 & 1 & \\hdots & 1\\\\ (1) & (1) & 1 & \\hdots & 1\\\\ & & & \\vdots & \\\\ (1) & (1) & (1) & \\hdots & 1 \\\\ (1) & (1) & (1) & \\hdots & (1), \\end{array}$<\/p>\n
where I’ve put parentheses on the $latex 1$’s from the second, copied triangle. By analogy, the sum of the integers from the original sequence is equal to half the area of a rectangle that is characterized by this matrix of ones—that is, a rectangle of height $latex n+1$ and width $latex n$. As such, $latex S = n(n+1)\/2$. \u00a0This comes up, e.g., in the asymptotic approximation for the Wilcoxon signed rank test (link<\/a>).<\/p>\n","protected":false},"excerpt":{"rendered":" Math is often about finding the right analogy, often a spatial analogy. \u00a0Suppose you want to compute the sum of integers, $latex S = 1+2+3+ \\hdots + N$. Consider decomposing the sum as, $latex \\begin{array}{cccccc}S = & 1 & + 1 & + 1 & + \\hdots & + 1\\\\ & & + 1 & … <\/p>\n