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1x1,1x2, ..., 1xn tiles in a nxn board, red n (n + 1)/2 cells

Source: 2020 Dutch IMO TST 1.3

November 21, 2020
combinatoricsTilingtiles

Problem Statement

For a positive integer nn, we consider an n×nn \times n board and tiles with dimensions 1×1,1×2,...,1×n1 \times 1, 1 \times 2, ..., 1 \times n. In how many ways exactly can 12n(n+1)\frac12 n (n + 1) cells of the board are colored red, so that the red squares can all be covered by placing the nn tiles all horizontally, but also by placing all nn tiles vertically?
Two colorings that are not identical, but by rotation or reflection from the board into each other count as different.
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