Sequences counting and ranking integer partitions by the differences of their successive parts.
by Gus Wiseman, May 01 2019

The differences dif(y) (first column) of an integer partition y of length k are given by dif(y)_i = y_{i + 1} - y_i for i < k. For example, dif(6,5,5,3,3,3) = (-1,0,-2,0,0).

The augmented differences aug(y) (second column) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The differences with 0 appended app(y) (third column) of an integer partition y of length k are given by app(y)_i = y_{i + 1} - y_i if i < k and app(y)_k = - y_k. For example, app(6,5,5,3,3,3) = (-1,0,-2,0,0,-3).

The zeroth differences of an integer partition y are the the parts of the partition itself, in the usual weakly decreasing order, while the k-th differences for k > 0 are the differences of the (k-1)-th differences. For a condition on the differences of all degrees (fourth column) of an integer partition y to be met, the condition must hold independently for the k-th differences of y for each k >= 0.


Number of partitions of n:

                        dif:     aug:     app:     all:
             distinct:  A325325  A325349  A325324  A325468
                equal:  A049988  A129654  A007862  A000005
    weakly increasing:  A240026  A325356  A007294  A325354
    weakly decreasing:  A320466  A325350  A320509  A325353
  strictly increasing:  A240027  A325357  A179269  A325391
  strictly decreasing:  A320470  A325358  A320510  A325393


Heinz numbers of partitions:

                        dif:     aug:     app:     all:
             distinct:  A325368  A325366  A325367  A325467
                equal:  A325328  A307824  A325327  A000961
    weakly increasing:  A325360  A325394  A325362  A325400
    weakly decreasing:  A325361  A325389  A325364  A325397
  strictly increasing:  A325456  A325395  A325460  A325398
  strictly decreasing:  A325457  A325396  A325461  A325399