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