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%I #25 Feb 23 2021 10:08:15
%S 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,
%T 78,102,92,123,123,146,148,191,181,228,233,280,283,348,350,420,437,
%U 518,523,616,641,727,774,884,911,1038,1102,1240,1292,1463,1530,1715,1861,2002
%N 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.
%C 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
%H Fausto A. C. Cariboni, <a href="/A320348/b320348.txt">Table of n, a(n) for n = 1..500</a> (terms 1..100 from Seiichi Manyama)
%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e n = 9
%e [9] ********* a_1 = 9.
%e ooooooooo
%e ------------------------------------
%e [8, 1] * a_2 = 1.
%e *******o a_1 - a_2 = 7.
%e oooooooo
%e ------------------------------------
%e [7, 2] ** a_2 = 2.
%e *****oo a_1 - a_2 = 5.
%e ooooooo
%e ------------------------------------
%e [5, 4] **** a_2 = 4.
%e *oooo a_1 - a_2 = 1.
%e ooooo
%e ------------------------------------
%e a(9) = 4.
%e From _Gus Wiseman_, May 04 2019: (Start)
%e 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.
%e (1) (2) (3) (4) (5) (6) (7) (8) (9) (A) (B)
%e (31) (32) (51) (43) (53) (54) (64) (65)
%e (41) (52) (62) (72) (73) (74)
%e (61) (71) (81) (82) (83)
%e (91) (92)
%e (631) (A1)
%e (632)
%e (641)
%e (731)
%e 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.
%e 1 11 111 211 221 21111 2221 22211 22221 222211
%e 1111 2111 111111 22111 221111 2211111 322111
%e 11111 211111 2111111 21111111 2221111
%e 1111111 11111111 111111111 22111111
%e 211111111
%e 1111111111
%e 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.
%e (1) (2) (3) (4) (5) (6) (7) (8) (9) (A)
%e (211) (221) (411) (322) (332) (441) (433)
%e (311) (331) (422) (522) (442)
%e (511) (611) (711) (622)
%e (811)
%e (322111)
%e (End)
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Differences[Append[#,0]]&]],{n,30}] (* _Gus Wiseman_, May 04 2019 *)
%Y Cf. A000009, A320347.
%Y Cf. A007294, A007862, A048767, A098859, A179269, A320509, A320510, A325324, A325325, A325349, A325367, A325404, A325468.
%K nonn
%O 1,4
%A _Seiichi Manyama_, Oct 11 2018
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