A325356 Number of integer partitions of n whose augmented differences are weakly increasing.

1, 1, 2, 2, 3, 3, 4, 3, 6, 5, 5, 6, 8, 6, 10, 9, 8, 10, 13, 10, 15, 14, 13, 15, 21, 15, 19, 21, 20, 25, 25, 20, 31, 30, 30, 32, 35, 28, 40, 44, 36, 42, 50, 43, 54, 53, 49, 57, 67, 58, 68, 66, 66, 78, 84, 71, 86, 92, 82, 99, 109

Offset

0,3

Comments

The augmented differences aug(y) 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 Heinz numbers of these partitions are given by A325394.

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1500

Example

The a(1) = 1 through a(8) = 6 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (1111)  (11111)  (222)     (1111111)  (53)
                                     (111111)             (332)
                                                          (2222)
                                                          (11111111)
For example, the augmented differences of (6,6,5,3) are (1,2,3,3), which are weakly increasing, so (6,6,5,3) is counted under a(20).

Mathematica

aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], OrderedQ[aug[#]]&]], {n, 0, 30}]

Crossrefs

Cf. A000837, A007294, A049988, A098859, A325350, A325351, A325354, A325357, A325358, A325360, A325394.

Sequence in context: A027833 A110676 A117171 * A304706 A084054 A106747

Adjacent sequences: A325353 A325354 A325355 * A325357 A325358 A325359

Keyword

nonn

Author

Gus Wiseman, Apr 23 2019

Status

approved