A342095 - OEIS

A342095

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

%I #11 Jan 29 2022 04:20:36

%S 1,1,2,1,2,3,3,2,4,4,6,7,6,8,10,9,13,16,17,20,25,26,29,36,40,45,55,61,

%T 69,81,90,103,119,132,154,176,196,225,254,282,323,364,403,458,519,582,

%U 655,735,822,922,1035,1153,1290,1441,1600,1788,1997,2217,2468

%N Number of strict 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="/A342095/b342095.txt">Table of n, a(n) for n = 1..400</a>

%e The a(1) = 1 through a(15) = 10 partitions (A..F = 10..15):

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

%e 21 32 42 43 53 54 64 65 75 76 86 87

%e 321 421 63 532 74 84 85 95 96

%e 432 4321 542 543 643 653 A5

%e 632 642 742 743 654

%e 5321 5421 6421 842 753

%e 6321 5432 843

%e 7421 6432

%e 8421

%e 54321

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

%Y The reciprocal version (no adjacent parts having quotient < 2) is A000929.

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

%Y The multiplicative version is A342085 or A337135.

%Y The non-strict version is A342094.

%Y The non-strict version without quotients of 2 exactly is A342096.

%Y The version without quotients of 2 exactly is 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. A001055, A001227, A003242, A056239, A167606, A342083, A342084.

%K nonn

%O 1,3

%A _Gus Wiseman_, Mar 02 2021