A342097 Number of strict integer partitions of n with no adjacent parts having quotient >= 2.

1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 3, 3, 4, 6, 6, 7, 8, 8, 9, 11, 13, 15, 18, 20, 24, 25, 29, 32, 39, 42, 48, 54, 63, 72, 81, 89, 102, 116, 132, 147, 165, 187, 210, 238, 264, 296, 329, 371, 414, 465, 516, 580, 644, 722, 803, 897, 994, 1108, 1229, 1367, 1512, 1678

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 its negated first-differences.

Examples

			The a(1) = 1 through a(16) = 7 partitions (A..G = 10..16):
  1  2  3  4  5   6  7   8   9    A    B   C    D    E     F     G
              32     43  53  54   64   65  75   76   86    87    97
                             432  532  74  543  85   95    96    A6
                                                643  653   654   754
                                                     743   753   853
                                                     5432  6432  6532
                                                                 7432
		

Links

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

Crossrefs

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

The multiplicative version is A342083 or A342084.

The non-strict version allowing quotients of 2 exactly is A342094.

The version allowing quotients of 2 exactly is A342095.

The non-strict version is A342096.

The reciprocal version is A342098.

A000009 counts strict partitions.

A000929 counts partitions with no adjacent parts having quotient < 2.

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

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

Cf. A027193, A001055, A001227, A003242, A167606, A337135, A342085.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Thread[Differences[-#]