Generate basis elements of the Steenrod algebra
05AB1E, 16 12 bytes

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Åœíʒx¦@P}ʒÆ@


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Explanation

Åœ              # integer partitions
  í             # reverse each
   ʒ    }       # filter, keep only elements true under:
    x           # each element doubled
     ¦          # remove the first element in the doubled list
      @         # compare with greater than or equal with the non-doubled
       P        # product
         ʒ      # filter, keep only elements true under:
          Æ     # reduce by subtraction
           @    # compare with greater than or equal to second input