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

Offset: 1

Author

Seiichi Manyama, Oct 11 2018

Keywords

• nonn

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

Examples

			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)
		

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.

Crossrefs

Cf. A000009, A320347.

Cf. A007294, A007862, A048767, A098859, A179269, A320509, A320510, A325324, A325325, A325349, A325367, A325404, A325468.

Programs

Mathematica

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