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

Author

Gus Wiseman, Mar 20 2021

Keywords

• nonn

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).

Examples

			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
		

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 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.

Programs

Mathematica

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