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Number of integer partitions of n with no difference -2.
27

%I #10 Jan 27 2022 20:46:41

%S 1,1,2,3,4,6,9,12,16,24,30,40,54,69,89,118,146,187,239,297,372,468,

%T 575,711,880,1075,1314,1610,1947,2359,2864,3438,4135,4973,5936,7090,

%U 8466,10044,11922,14144,16698,19704,23249,27306,32071,37639,44019,51457,60113

%N Number of integer partitions of n with no difference -2.

%H Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.

%e The a(1) = 1 through a(7) = 12 partitions:

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

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

%e (111) (211) (41) (51) (52)

%e (1111) (221) (222) (61)

%e (2111) (321) (322)

%e (11111) (411) (511)

%e (2211) (2221)

%e (21111) (3211)

%e (111111) (4111)

%e (22111)

%e (211111)

%e (1111111)

%t Table[Length[Select[IntegerPartitions[n],FreeQ[Differences[#],-2]&]],{n,0,30}]

%Y Heinz number rankings are in parentheses below.

%Y The version for no difference 0 is A000009.

%Y The version for subsets of prescribed maximum is A005314.

%Y The version for all differences < -2 is A025157, non-strict A116932.

%Y The version for all differences > -2 is A034296, strict A001227.

%Y The opposite version is A072670.

%Y The version for no difference -1 is A116931 (A319630), strict A003114.

%Y The multiplicative version is A350837 (A350838), strict A350840.

%Y The strict case is A350844.

%Y The complement for quotients is counted by A350846 (A350845).

%Y A000041 = integer partitions.

%Y A027187 = partitions of even length.

%Y A027193 = partitions of odd length (A026424).

%Y A323092 = double-free partitions (A320340), strict A120641.

%Y A325534 = separable partitions (A335433).

%Y A325535 = inseparable partitions (A335448).

%Y A350839 = partitions with a gap and conjugate gap (A350841).

%Y Cf. A000070, A000929, A001511, A003242, A007359, A018819, A040039, A045690, A045691, A101417, A154402, A323093.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jan 20 2022