login
Number of integer partitions of n with no adjacent parts having quotient > 2.
38

%I #17 Feb 01 2022 05:12:22

%S 1,2,3,4,5,8,9,13,16,21,27,37,44,59,75,94,117,153,186,238,296,369,458,

%T 573,701,870,1068,1312,1601,1964,2384,2907,3523,4270,5159,6235,7491,

%U 9021,10819,12964,15494,18517,22049,26260,31195,37020,43851,51906,61290

%N Number of integer partitions of n with no adjacent parts having quotient > 2.

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

%H Fausto A. C. Cariboni, <a href="/A342094/b342094.txt">Table of n, a(n) for n = 1..250</a>

%e The a(1) = 1 through a(8) = 13 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (21) (22) (32) (33) (43) (44)

%e (111) (211) (221) (42) (322) (53)

%e (1111) (2111) (222) (421) (332)

%e (11111) (321) (2221) (422)

%e (2211) (3211) (2222)

%e (21111) (22111) (3221)

%e (111111) (211111) (4211)

%e (1111111) (22211)

%e (32111)

%e (221111)

%e (2111111)

%e (11111111)

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

%Y The version with no adjacent parts having quotient < 2 is A000929.

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

%Y A strong multiplicative version is A342083 or A342084.

%Y The multiplicative version is A342085, with reciprocal version A337135.

%Y The strict case is A342095.

%Y The version with all adjacent parts having quotient < 2 is A342096, with strict case A342097.

%Y The version with all adjacent parts having quotient > 2 is A342098.

%Y The Heinz numbers of these partitions are listed by A342191.

%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 A038548 counts inferior (or superior) divisors, listed by A161906.

%Y A161908 lists superior prime divisors.

%Y Cf. A001055, A001227, A003242, A027193, A167606, A178470, A253784, A341591.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 02 2021