A332283 - OEIS

A332283

Number of integer partitions of n whose first differences (assuming the last part is zero) are unimodal.

%I #12 Feb 16 2025 08:33:59

%S 1,1,2,3,5,7,10,13,18,24,30,38,49,59,73,90,108,129,159,184,216,258,

%T 298,347,410,466,538,626,707,807,931,1043,1181,1351,1506,1691,1924,

%U 2132,2382,2688,2971,3300,3704,4073,4500,5021,5510,6065,6740,7362,8078

%N Number of integer partitions of n whose first differences (assuming the last part is zero) are unimodal.

%C First differs from A000041 at a(6) = 10, A000041(6) = 11.

%C A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

%H Fausto A. C. Cariboni, <a href="/A332283/b332283.txt">Table of n, a(n) for n = 0..400</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.

%e The a(1) = 1 through a(7) = 13 partitions:

%e (1) (2) (3) (4) (5) (6) (7)

%e (11) (21) (22) (32) (33) (43)

%e (111) (31) (41) (42) (52)

%e (211) (221) (51) (61)

%e (1111) (311) (222) (322)

%e (2111) (321) (421)

%e (11111) (411) (511)

%e (3111) (2221)

%e (21111) (3211)

%e (111111) (4111)

%e (31111)

%e (211111)

%e (1111111)

%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];

%t Table[Length[Select[IntegerPartitions[n],unimodQ[Differences[Append[#,0]]]&]],{n,0,30}]

%Y Unimodal compositions are A001523.

%Y Unimodal normal sequences appear to be A007052.

%Y Partitions with unimodal run-lengths are A332280.

%Y Heinz numbers of partitions with non-unimodal run-lengths are A332282.

%Y The complement is counted by A332284.

%Y The strict case is A332285.

%Y Heinz numbers of partitions not in this class are A332287.

%Y Cf. A025065, A072706, A115981, A227038, A332288, A332577, A332638, A332642.

%K nonn,changed

%O 0,3

%A _Gus Wiseman_, Feb 19 2020