A320348 - OEIS

A320348

Number of partition into distinct parts (a_1, a_2, ... , a_m) (a_1 > a_2 > ... > a_m and Sum_{k=1..m} a_k = n) such that a1 - a2, a2 - a_3, ... , a_{m-1} - a_m, a_m are different.

1, 1, 1, 2, 3, 2, 4, 4, 4, 6, 9, 7, 13, 12, 13, 16, 22, 17, 28, 28, 31, 36, 50, 45, 63, 62, 74, 78, 102, 92, 123, 123, 146, 148, 191, 181, 228, 233, 280, 283, 348, 350, 420, 437, 518, 523, 616, 641, 727, 774, 884, 911, 1038, 1102, 1240, 1292, 1463, 1530, 1715, 1861, 2002

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OFFSET

1,4

COMMENTS

Also the number of integer partitions of n whose parts cover an initial interval of positive integers with distinct multiplicities. Also the number of integer partitions of n whose multiplicities cover an initial interval of positive integers and are distinct (see A048767 for a bijection). - Gus Wiseman, May 04 2019

LINKS

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..500 (terms 1..100 from Seiichi Manyama)

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

EXAMPLE

n = 9

[9] ********* a_1 = 9.

ooooooooo

------------------------------------

[8, 1] * a_2 = 1.

*******o a_1 - a_2 = 7.

oooooooo

------------------------------------

[7, 2] ** a_2 = 2.

*****oo a_1 - a_2 = 5.

ooooooo

------------------------------------

[5, 4] **** a_2 = 4.

*oooo a_1 - a_2 = 1.

ooooo

------------------------------------

a(9) = 4.

From Gus Wiseman, May 04 2019: (Start)

The a(1) = 1 through a(11) = 9 strict partitions with distinct differences (where the last part is taken to be 0) are the following (A = 10, B = 11). The Heinz numbers of these partitions are given by A325388.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (A) (B)

(31) (32) (51) (43) (53) (54) (64) (65)

(41) (52) (62) (72) (73) (74)

(61) (71) (81) (82) (83)

(91) (92)

(631) (A1)

(632)

(641)

(731)

The a(1) = 1 through a(10) = 6 partitions covering an initial interval of positive integers with distinct multiplicities are the following. The Heinz numbers of these partitions are given by A325326.

1 11 111 211 221 21111 2221 22211 22221 222211

1111 2111 111111 22111 221111 2211111 322111

11111 211111 2111111 21111111 2221111

1111111 11111111 111111111 22111111

211111111

1111111111

The a(1) = 1 through a(10) = 6 partitions whose multiplicities cover an initial interval of positive integers and are distinct are the following (A = 10). The Heinz numbers of these partitions are given by A325337.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (A)

(211) (221) (411) (322) (332) (441) (433)

(311) (331) (422) (522) (442)

(511) (611) (711) (622)

(811)

(322111)

(End)

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&UnsameQ@@Differences[Append[#, 0]]&]], {n, 30}] (* Gus Wiseman, May 04 2019 *)

CROSSREFS

Cf. A000009, A320347.