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 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. 21
 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 (list; graph; refs; listen; history; text; internal format)
 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 Seiichi Manyama, Table of n, a(n) for n = 1..100 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. Cf. A007294, A007862, A048767, A098859, A179269, A320509, A320510, A325324, A325325, A325349, A325367, A325404, A325468. Sequence in context: A249870 A325404 A324750 * A145394 A179806 A182762 Adjacent sequences:  A320345 A320346 A320347 * A320349 A320350 A320351 KEYWORD nonn AUTHOR Seiichi Manyama, Oct 11 2018 STATUS approved

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Last modified December 12 22:06 EST 2019. Contains 329963 sequences. (Running on oeis4.)