A342095 Number of strict integer partitions of n with no adjacent parts having quotient > 2.

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, 69, 81, 90, 103, 119, 132, 154, 176, 196, 225, 254, 282, 323, 364, 403, 458, 519, 582, 655, 735, 822, 922, 1035, 1153, 1290, 1441, 1600, 1788, 1997, 2217, 2468

Offset

1,3

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..400

Example

The a(1) = 1 through a(15) = 10 partitions (A..F = 10..15):
  1  2  3   4  5   6    7    8   9    A     B     C     D     E     F
        21     32  42   43   53  54   64    65    75    76    86    87
                   321  421      63   532   74    84    85    95    96
                                 432  4321  542   543   643   653   A5
                                            632   642   742   743   654
                                            5321  5421  6421  842   753
                                                  6321        5432  843
                                                              7421  6432
                                                                    8421
                                                                    54321

Mathematica

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

Crossrefs

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

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

The multiplicative version is A342085 or A337135.

The non-strict version is A342094.

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

The version without quotients of 2 exactly is A342097.

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.

Cf. A001055, A001227, A003242, A056239, A167606, A342083, A342084.

Sequence in context: A057475 A024376 A230128 * A340828 A123265 A104345

Adjacent sequences: A342092 A342093 A342094 * A342096 A342097 A342098

Keyword

nonn

Author

Gus Wiseman, Mar 02 2021

Status

approved