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Number of strict integer partitions of n with no difference -2.
16

%I #9 Jan 25 2022 10:26:18

%S 1,1,1,2,1,3,3,4,4,7,7,8,11,12,15,18,21,23,31,32,40,45,54,59,73,78,94,

%T 106,122,136,161,177,203,231,259,293,334,372,417,476,525,592,663,742,

%U 821,931,1020,1147,1271,1416,1558,1752,1916,2137,2357,2613,2867

%N Number of strict 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(12) = 11 partitions (A..C = 10..12):

%e 1 2 3 4 5 6 7 8 9 A B C

%e 21 32 51 43 62 54 73 65 84

%e 41 321 52 71 63 82 74 93

%e 61 521 72 91 83 A2

%e 81 541 92 B1

%e 432 721 A1 543

%e 621 4321 632 651

%e 821 732

%e 741

%e 921

%e 6321

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

%Y The version for no difference 0 is A000009.

%Y The version for no difference > -2 is A001227, non-strict A034296.

%Y The version for no difference -1 is A003114 (A325160).

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

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

%Y The opposite version is A072670.

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

%Y The non-strict version is A350842.

%Y A000041 counts integer partitions.

%Y A027187 counts partitions of even length.

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

%Y A116931 counts partitions with no difference -1 (A319630).

%Y A323092 counts double-free integer partitions (A320340) strict A120641.

%Y A325534 counts separable partitions (A335433).

%Y A325535 counts inseparable partitions (A335448).

%Y Cf. A000929, A003000, A018819, A040039, A045690, A045691, A154402, A303362, A323094, A342095, A342097.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jan 21 2022