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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?

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The general case is Young tableaux. They generalize your rectangular arrangements. They're useful in the representation theory of $S_n$, where one usually considers them with "unordered rows", and sometimes also with "unordered columns".

The operation of taking a $3\times 4$ table and turning it into a $4\times 3$ table corresponds to moving to the "transposed" or "conjugate" tableau (or rather Ferrers diagram, which is the tableau without the numbers), which has a simple interpretation in terms of the resulting representation (it is multiplied by the sign representation).

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To more directly answer the question, the subgroups that don't change a row sum or that don't change a column sum are called the Young subgroups of the symmetric group. The non-commutativity is pretty directly the non-commutativity of permutation multiplication. –  Jack Schmidt Feb 15 '11 at 14:42

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