Coarea on DyckPath
The complement of the area within the staircase: binomial(n, 2) less the area.
Zero for the highest path, UU...DD, whose area is binomial(n, 2).
Called as CombinatorialStat(x, "Coarea"). The head's page has its examples.
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Distribution
Over DyckPaths at each size, tallied here in your browser; sizes past 10 are not offered.