login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A325356 Number of integer partitions of n whose augmented differences are weakly increasing. 13
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..60.

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 26 21:46 EDT 2020. Contains 337377 sequences. (Running on oeis4.)