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

Author

Gus Wiseman, Mar 02 2021

Keywords

• nonn

Comments

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

Examples

			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
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 1..400

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.

Programs

Mathematica

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