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Number of (necessarily strict) integer partitions of n with all adjacent parts having quotients > 2.
42

%I #9 Jan 26 2022 08:28:05

%S 1,1,1,2,2,2,3,3,3,4,5,5,6,7,7,8,9,10,11,12,13,14,15,16,18,20,21,23,

%T 25,26,28,31,33,35,38,40,42,45,48,51,55,58,61,65,68,72,77,81,85,90,94,

%U 98,104,109,114,121,127,132,139,146

%N Number of (necessarily strict) integer partitions of n with all adjacent parts having quotients > 2.

%C The decapitation of such a partition (delete the greatest part) is term-wise less than its negated first-differences.

%H Fausto A. C. Cariboni, <a href="/A342098/b342098.txt">Table of n, a(n) for n = 1..3000</a>

%e The a(1) = 1 through a(16) = 8 partitions (A..G = 10..16):

%e 1 2 3 4 5 6 7 8 9 A B C D E F G

%e 31 41 51 52 62 72 73 83 93 94 A4 B4 B5

%e 61 71 81 82 92 A2 A3 B3 C3 C4

%e 91 A1 B1 B2 C2 D2 D3

%e 731 831 C1 D1 E1 E2

%e 931 941 A41 F1

%e A31 B31 B41

%e C31

%t Table[Length[Select[IntegerPartitions[n],And@@Thread[Differences[-#]>Rest[#]]&]],{n,30}]

%Y The version allowing equality is A000929.

%Y The case of equality (all adjacent parts having quotient 2) is A154402.

%Y The multiplicative version is A342083.

%Y The version with all quotients <= 2 is A342094 (strict: A342095).

%Y The version with all quotients < 2 is A342096 (strict: A342097).

%Y A000009 counts strict partitions.

%Y A003114 counts partitions with adjacent parts differing by more than 1.

%Y A034296 counts partitions with adjacent parts differing by at most 1.

%Y Cf. A027193, A001055, A001227, A003242, A167606, A178470, A337135, A342084, A342085.

%K nonn

%O 1,4

%A _Gus Wiseman_, Mar 04 2021