So the allowable orderings of the (N-1)*N/2 entries in the strict lower triangle of an NxN matrix (strict lower triangle only) according to my two criteria 1. 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. 2. the ordering must be such that the integers increase going upward within each column. ("vertical monotonicity.") They are as follows: N=2: 1 N=3: 2 1 3 N=4: 4 2 5 1 3 6 3 2 5 1 4 6 N=5: 7 4 8 2 5 9 1 3 6 10 6 4 8 2 5 9 1 3 7 10 6 4 7 2 5 9 1 3 8 10 5 4 8 2 6 9 1 3 7 10 5 4 7 2 6 9 1 3 8 10 7 3 8 2 5 9 1 4 6 10 6 3 8 2 5 9 1 4 7 10 6 3 7 2 5 9 1 4 8 10 5 3 8 2 6 9 1 4 7 10 5 3 7 2 6 9 1 4 8 10 4 3 8 2 6 9 1 5 7 10 4 3 7 2 6 9 1 5 8 10 And the sequence giving the numbers of allowable orderings begins (for N=1,2,3,4,5,6,7): 1, 1, 1, 2, 12, 286, 33592 which agrees with OEIS https://oeis.org/A003121 "strict ballot numbers". Formula known and stated there, and indeed grows SuperExponentially(N^2) as I had originally conjectured. Well, isn't that nifty. So for any NxN orthogonal matrix, there are at least F(N) ways to write it as a product of N*(N-1)/2 terms, each a 2x2 rotation for adjacent coordinates, where F(N) is the sequence above... Indeed more strongly, if the matrix is generic there are exactly F(N) ways to multiply the orthog by 2x2s on the left, N*(N-1)/2 of them in all, each involving adjacent coordinates, such that the strict lower triangle gets zeroed one entry at a time, never unzeroing. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)