A342516 Number of strict integer partitions of n with weakly increasing first quotients.

1, 1, 1, 2, 2, 3, 3, 5, 5, 6, 7, 8, 8, 11, 12, 14, 15, 17, 17, 21, 22, 26, 29, 31, 32, 35, 38, 42, 45, 48, 51, 58, 59, 63, 70, 76, 80, 88, 94, 98, 105, 113, 121, 129, 133, 143, 153, 159, 166, 183, 189, 195, 210, 221, 231, 248, 262, 273, 284, 298, 312

Offset

0,4

Comments

Also called log-concave-up strict partitions.

Also the number of reversed strict integer partitions of n with weakly 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).

Links

Table of n, a(n) for n=0..60.

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.

Example

The partition (6,3,2,1) has first quotients (1/2,2/3,1/2) so is not counted under a(12), even though the first differences (-3,-1,-1) are weakly increasing.
The a(1) = 1 through a(13) = 11 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
                              421   521   81    91    92    A2    A3
                                          621   532   A1    B1    B2
                                                721   632   732   C1
                                                      821   921   643
                                                                  832
                                                                  931
                                                                  A21

Mathematica

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

Crossrefs

The version for differences instead of quotients is A179255.

The non-strict ordered version is A342492.

The non-strict version is A342497 (ranking: A342523).

The strictly increasing version is A342517.

The weakly decreasing version is A342519.

A000041 counts partitions (strict: A000009).

A000929 counts partitions with all adjacent parts x >= 2y.

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

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

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

A342094 counts partitions with all adjacent parts x <= 2y (strict: A342095).

Cf. A000005, A003114, A003242, A005117, A057567, A067824, A238710, A253249, A318991, A318992, A342528.

Sequence in context: A178503 A211275 A240542 * A325391 A179254 A304430

Adjacent sequences: A342513 A342514 A342515 * A342517 A342518 A342519

Keyword

nonn

Author

Gus Wiseman, Mar 20 2021

Status

approved