Sequences counting and ranking partitions and compositions by their differences and quotients.

By Gus Wiseman
Last Updated Mar 15 2021.


- The first differences of a sequence are defined as if it were increasing, so for example the first differences of (6,3,1) are (-3,-2).
- The first quotients of a sequence are defined as if it were an increasing chain of divisors, so for example the first quotients of (6,3,1) are (1/2,1/3).
- "dstnct" means all parts are different.
- "equal" means all parts are equal.
- "wkinc" means the parts are weakly increasing.
- "wkdec" means the parts are weakly decreasing.
- "strinc" means the parts are strictly increasing.
- "strdec" means the parts are strictly decreasing.
- The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.


First differences
                      dstnct   equal    wkinc    wkdec    strinc   strdec
                     ------------------------------------------------------
         Partitions:  A325325  A049988  A240026  A320466  A240027  A320470
      Heinz numbers:  A325368  A325328  A325360  A325361  A325456  A325457
       Compositions:  A325545  A175342  A325546  A070211  A325547  A325548
  Strict partitions:  A320347  A049980  A179255  A320382  A179254  A320388


First quotients
                      dstnct   equal    wkinc    wkdec    strinc   strdec
                     ------------------------------------------------------
         Partitions:  A342514  A342496  A342497  A342513  A342498  A342499
      Heinz numbers:  A342521  A342522  A342523  A342526  A342524  A342525
       Compositions:  A342529  A342495  A342492  A069916  A342493  A342494
  Strict partitions:  A342520  A342515  A342516  A342519  A342517  A342518





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