A342094 Number of integer partitions of n with no adjacent parts having quotient > 2.

1, 2, 3, 4, 5, 8, 9, 13, 16, 21, 27, 37, 44, 59, 75, 94, 117, 153, 186, 238, 296, 369, 458, 573, 701, 870, 1068, 1312, 1601, 1964, 2384, 2907, 3523, 4270, 5159, 6235, 7491, 9021, 10819, 12964, 15494, 18517, 22049, 26260, 31195, 37020, 43851, 51906, 61290

Offset

1,2

Comments

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

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..250

Example

The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (211)   (221)    (42)      (322)      (53)
                    (1111)  (2111)   (222)     (421)      (332)
                            (11111)  (321)     (2221)     (422)
                                     (2211)    (3211)     (2222)
                                     (21111)   (22111)    (3221)
                                     (111111)  (211111)   (4211)
                                               (1111111)  (22211)
                                                          (32111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)

Mathematica

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

Crossrefs

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

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

A strong multiplicative version is A342083 or A342084.

The multiplicative version is A342085, with reciprocal version A337135.

The strict case is A342095.

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

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

The Heinz numbers of these partitions are listed by A342191.

A000009 counts strict partitions.

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

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

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

A161908 lists superior prime divisors.

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

Sequence in context: A063678 A005424 A105317 * A094103 A349412 A317082

Adjacent sequences: A342091 A342092 A342093 * A342095 A342096 A342097

Keyword

nonn

Author

Gus Wiseman, Mar 02 2021

Status

approved