A342519 Number of strict integer partitions of n with weakly decreasing first quotients.

1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 9, 12, 14, 15, 18, 18, 21, 25, 29, 32, 38, 40, 44, 51, 57, 61, 66, 73, 77, 89, 97, 104, 115, 124, 135, 147, 160, 174, 193, 206, 218, 238, 254, 272, 293, 313, 331, 353, 381, 408, 436, 468, 499, 532, 569, 610, 651, 694, 735, 783

Offset

0,4

Comments

Also called log-concave-down strict partitions.

Also the number of reversed strict partitions of n with weakly decreasing 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 strict partition (10,7,4,2,1) has first quotients (7/10,4/7,1/2,1/2) so is counted under a(24), even though the first differences (-3,-3,-2,-1) are weakly increasing.
The a(1) = 1 through a(13) = 14 strict 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
                         321   61    71    72    82     83    93     94
                               421   431   81    91     92    A2     A3
                                           432   541    A1    B1     B2
                                           531   631    542   543    C1
                                                 4321   641   642    652
                                                        731   651    742
                                                              741    751
                                                              831    841
                                                              5421   931
                                                                     5431
                                                                     6421

Mathematica

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

Crossrefs

The non-strict ordered version is A069916.

The version for differences instead of quotients is A320382.

The non-strict version is A342513 (ranking: A342526).

The weakly increasing version is A342516.

The strictly decreasing version is A342518.

A000005 counts constant partitions.

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).

A057567 counts strict chains of divisors with weakly increasing quotients.

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

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

A342528 counts compositions with alternately weakly increasing parts.

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

Sequence in context: A351007 A307779 A165684 * A307780 A340276 A340275

Adjacent sequences: A342516 A342517 A342518 * A342520 A342521 A342522

Keyword

nonn

Author

Gus Wiseman, Mar 20 2021

Status

approved