A342518 Number of strict integer partitions of n with strictly decreasing first quotients.

1, 1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 9, 11, 12, 13, 17, 18, 21, 24, 28, 30, 34, 37, 41, 47, 52, 56, 63, 68, 72, 83, 89, 99, 108, 117, 128, 139, 149, 163, 179, 189, 203, 217, 233, 250, 272, 289, 305, 329, 355, 381, 410, 438, 471, 505, 540, 571, 607, 645, 683, 726

Offset: 0

Author

Gus Wiseman, Mar 20 2021

Keywords

• nonn

Comments

Also the number of reversed strict integer partitions of n with strictly 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 (12,10,6,3,1) has first quotients (5/6,3/5,1/2,1/3) so is counted under a(32), even though the differences (-2,-4,-3,-2) are not strictly decreasing.
The a(1) = 1 through a(13) = 12 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
                                    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
                                                                   5431
		

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 A320388.

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

The non-strict ordered version is A342494.

The non-strict version is A342499 (ranking: A342525).

The strictly increasing version is A342517.

The weakly decreasing version is A342519.

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@@#&&Greater@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]