A342517 Number of strict integer partitions of n with strictly increasing first quotients.

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 8, 10, 11, 13, 14, 16, 16, 19, 21, 23, 27, 29, 31, 34, 36, 40, 43, 47, 49, 53, 56, 59, 66, 71, 75, 81, 86, 89, 97, 104, 110, 119, 123, 132, 143, 148, 156, 168, 177, 184, 198, 209, 218, 232, 246, 257, 269, 282, 294

Offset: 0

Author

Gus Wiseman, Mar 20 2021

Keywords

• nonn

Comments

Also the number of reversed strict partitions of n with strictly increasing first quotients.

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

Examples

			The partition (14,8,5,3,2) has first quotients (4/7,5/8,3/5,2/3) so is not counted under a(32), even though the differences (-6,-3,-2,-1) are strictly increasing.
The a(1) = 1 through a(13) = 10 partitions (A..D = 10..13):
  1   2   3    4    5    6    7    8     9     A     B     C     D
          21   31   32   42   43   53    54    64    65    75    76
                    41   51   52   62    63    73    74    84    85
                              61   71    72    82    83    93    94
                                   521   81    91    92    A2    A3
                                         621   532   A1    B1    B2
                                               721   632   732   C1
                                                     821   921   643
                                                                 832
                                                                 A21
		

Links

• Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
• Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
• Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

Crossrefs

The version for differences instead of quotients is A179254.

The version for chains of divisors is A342086 (non-strict: A057567).

The non-strict ordered version is A342493.

The non-strict version is A342498 (ranking: A342524).

The weakly increasing version is A342516.

The strictly decreasing version is A342518.

A000041 counts partitions (strict: A000009).

A001055 counts factorizations (strict: A045778, ordered: A074206).

A003238 counts chains of divisors summing to n - 1 (strict: A122651).

A045690 counts sets with maximum n with all adjacent elements y < 2x.

A167865 counts strict chains of divisors > 1 summing to n.

A342096 counts partitions with all adjacent parts x < 2y (strict: A342097).

A342098 counts (strict) partitions with all adjacent parts x > 2y.

Cf. A000005, A003114, A003242, A005117, A018819, A067824, A238710, A253249, A318991, A318992.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Less@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]