NILPSNevada International License Plate Society (license plate hobbyists)
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(In the NILP picture, each row has some number of vertical steps followed by some number of diagonal steps.)
It would seem reasonable to conjecture that the -1 of the ASM should correspond to all instances of a vertical step followed by a diagonal step as you go from left to right along a row of the NILP (or a 0 followed by a 1 as you go across a row of the boolean triangle).
So one might hope to begin a general bijection by determining the (n - 1) st row of the monotone triangle from the top row of the NILP (or the bottom row of the boolean triangle) by left-justifying all the vertical steps and then bijecting in the same way as in the permutation case.