I am working on a combinatorics library of mine, where Partitions and Compositions are also included. For unrestricted partitions, the definition of conjugate partition is fine. Conjugates are used in some transformations that swap between LEX and COLEX orderings.
Now I would like to use same principles for restricted partitions (eg with fixed number of parts). But in this case, the definition of conjugate/transpose partition produces partitions which are outside the class of partitions required (i.e no more with fixed num of parts, but with fixed greater part).
Ideally, I would expect "restricted conjugate" of $(5311)$ (LEX) to be $(3331)$ (COLEX), or alternatively the "restricted conjugate" of $(4222)$ (REVLEX) would be $(3331)$ (COLEX). Both are fine for what is needed, I can use any of the two (both cannot hold at the same time).
Is there an extension or alternative to the usual conjugate/transpose of a partition that maintains the restrictions applied (i.e maintains number of parts, maximum part, minimum part)?
Here's a modified conjugation that fixes the number of parts. I'll describe it in terms of Ferrers diagrams.
Conjugate the partition. If the result has $k$ too few rows (parts), turn the first $k$ columns into rows. For instance, with $k=4$, 3322 has conjugate 442 and then the mapping gives 3331---removing the leftmost column from 442 leaves 331 and that removed column became another 3. (This column-to-row operation is known as Bulgarian Solitaire.)
If the conjugate has too many rows, remove the partition from the $(k+1)$st row down, conjugate it, and append it to the remaining four rows. E.g., still with $k = 4$, 6411 has conjugate 422211 with 11 below the intended 4 rows; the conjugate of the small partition (guaranteed to have at most 4 parts) is 2 so the mapping gives 6222.
If the result of conjugation has the desired number of rows, you're done!
The map definitions may need tweaking and there are certainly details to check (for at least the two examples and a few small $n$ for which I worked everything out, this is an involution).
By the way, this made me think of Wilf partitions, where the (nonzero) multiplicities of its parts are all different. This class is closed by the involution that swaps parts and multiplicities, e.g., 422111 or $4^1 2^2 1^3$ becomes $3221111$ or $3^1 2^2 1^4$.