The question is equivalent to the following. How many ways can you fill in that triangle with the integers 1,2,3,...,(N-1)*N/2 in such a way that: if some cell contains integer K, then each cell on the same row containing an integer<K, must be accompanied by a "mate" -- the cell immediately above it -- also with contents <K. ?
--Sorry, it occurs to me there is another demand: the ordering must be such that the integers increase going upward within each column. ("vertical monotonicity.") This has a very major effect on the count of orderings and may totally destroy my heuristic estimate. Then we manually find the only two allowed orderings when N=4 are 4 2 5 1 3 6 3 2 5 1 4 6 OEIS finds 11820 sequences containing 1,1,1,2... (Perhaps for some reason horizontal monotonicity is also required? And/or monotonicity along the NW-SE diagonals? If so, the arguments for why will be indirect. The conditions on the orderings I have mentioned so far, are direct and presumably imply these further conditions if they are valid.) -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)