Suppose you have a set of twelve distinct objects, and you were interested in ways of arranging them into a 3 by 4 rectangle. You wonder: does it matter if I arrange them in 3 bunches of 4 or 4 bunches of 3?
Label them by primes. (I could, and probably should write $a_{2}$ in the rest of this post, but I don't feel like messing around with tables that much.) The numbers after the equal signs are row and column products.
2 3 5 7 11 13 17 19 23 29 31 37
Let's switch the 17 and the 19, again:
2 3 5 7 = 210 11 13 17 19 = 46189 23 29 31 37 = 1028859 = = = = 506 1131 2635 4921
If we switch the 17 and the 19:
2 3 5 7 = 210 11 13 19 17 = 46189 23 29 31 37 = 1028859 = = = = 506 1131 2945 4403
Now let's swap the 5 and 19:
2 3 19 7 = 798 11 13 5 17 = 12155 23 29 31 37 = 1028859 = = = = 506 1131 2945 4403
What's interesting about all of these is that each of these arrangements (factorizations if we're talking about numbers) has different uniquenesses about the rows and the columns
Look at the last two tables: In particular, the columns are pairwise the same, but the rows are pairwise different, so in this case it does make a difference whether we're talking about four bunches of three or three bunches of four, because there's a way of arranging it such that the four bunches of three are the same, but the three bunches of four are different. Is there an explanation of this "noncommutativity" in terms of subgroups of the symmetric group?