A342514 - OEIS

A342514

Number of integer partitions of n with distinct first quotients.

1, 1, 2, 2, 4, 5, 6, 8, 11, 14, 18, 24, 28, 35, 41, 52, 64, 81, 93, 115, 137, 157, 190, 225, 268, 313, 366, 430, 502, 587, 683, 790, 913, 1055, 1217, 1393, 1605, 1830, 2098, 2384, 2722, 3101, 3524, 4005, 4524, 5137, 5812, 6570, 7434, 8360, 9416, 10602, 11881

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OFFSET

0,3

COMMENTS

Also the number of reversed integer partitions of n with distinct 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..52.

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 (4,3,3,2,1) has first quotients (3/4,1,2/3,1/2) so is counted under a(13), but it has first differences (-1,0,-1,-1) so is not counted under A325325(13).

The a(1) = 1 through a(9) = 14 partitions:

(1) (2) (3) (4) (5) (6) (7) (8) (9)

(11) (21) (22) (32) (33) (43) (44) (54)

(31) (41) (42) (52) (53) (63)

(211) (221) (51) (61) (62) (72)

(311) (321) (322) (71) (81)

(411) (331) (332) (432)

(511) (422) (441)

(3211) (431) (522)

(521) (531)

(611) (621)

(3221) (711)

(3321)

(4311)

(5211)

MATHEMATICA

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

CROSSREFS

The version for differences instead of quotients is A325325.

The ordered version is A342529.

The strict case is A342520.

The Heinz numbers of these partitions are A342521.

A000005 counts constant partitions.

A000009 counts strict partitions.

A000041 counts partitions.

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.

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

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

Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

Sequence in context: A266745 A373219 A062436 * A238571 A325874 A293957

Adjacent sequences: A342511 A342512 A342513 * A342515 A342516 A342517

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 17 2021

STATUS

approved