I think we are mostly on the same page.

The first code snippet in your post is precisely what I was suggesting: recode the elements of the bit array in the 'x' column to easily discover the set of transformations (resulting in a permutation of the rows)) needed to rearrange the 'x' column in the desired form.

I think your second snippet of code executes what I was suggesting: take the permutation computed in the first snippet and apply it to every column of your bit array. I'm sorry for not making more effort to understand your code; my comments may very well be redundant.

My 'broader picture' comment was just suggesting that explicit twiddling of the bit array might be able to be bypassed. If I understand your $args array correctly, a value z at index i means that you have z columns with bitstring i. The permutation converts each bitstring into another bitstring, which has the effect of shuffling the elements of $args. For instance transposing bit rows 1 and 2 would turn bit pattern '3' into bit pattern '5', so if $args3 = z, then after the transposition $args5 = z. In effect z has been permuted from the '3' postion in the $args array to the '5' position. The same thing happens for all the other elements of $args. Thus permuting the bit rows in the bit array is tantamount to permuting the elements of $args. Mathematically, one would say that the automorphism of the bit array induces an automorphism the $args.

Whether manipulating the $args directly is faster than the bit twiddling depends on how slow the bit row copying is and, as you mention, speedups like memoization. I mostly made the comment for mathematical interest.