A320470 -id:A320470 - cp's OEIS frontend

A325325 Number of integer partitions of n with distinct differences between successive parts.

1, 1, 2, 2, 4, 5, 5, 8, 11, 12, 16, 22, 21, 30, 34, 42, 49, 64, 67, 87, 95, 117, 132, 160, 169, 207, 230, 274, 301, 360, 395, 463, 506, 602, 656, 762, 834, 960, 1042, 1220, 1311, 1505, 1643, 1859, 2000, 2341, 2491, 2827, 3083, 3464, 3747, 4302, 4561, 5154

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

The Heinz numbers of these partitions are given by A325368.

			The a(0) = 1 through a(9) = 12 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)  (411)  (322)  (71)    (81)
                                            (331)  (332)   (441)
                                            (421)  (422)   (522)
                                            (511)  (431)   (621)
                                                   (521)   (711)
                                                   (611)   (4221)
                                                   (4211)  (4311)
                                                           (5211)
For example, (5,2,1,1) has differences (-3,-1,0), which are distinct, so (5,2,1,1) is counted under a(9).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..400 (terms 0..123 from Alois P. Heinz)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A007294, A007862, A049988, A098859, A240026, A240027, A320348, A320466, A320470, A325324, A325349, A325352, A325368, A325404, A325468.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Differences[#]&]],{n,0,30}]

A240026 Number of partitions of n such that the successive differences of consecutive parts are nondecreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 12, 16, 21, 27, 32, 43, 50, 60, 75, 90, 103, 128, 146, 170, 203, 234, 264, 315, 355, 402, 467, 530, 589, 684, 764, 851, 969, 1083, 1195, 1360, 1504, 1659, 1863, 2063, 2258, 2531, 2779, 3039, 3379, 3709, 4032, 4474, 4880, 5304, 5846, 6373, 6891, 7578, 8227, 8894, 9727, 10550, 11357, 12405, 13404, 14419

Offset: 0

Joerg Arndt, Mar 31 2014

nonn

Partitions (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) <= p(k) - p(k-1) for all k >= 3.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). Then a(n) is the number of integer partitions of n whose differences are weakly increasing. The Heinz numbers of these partitions are given by A325360. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are weakly increasing, which is the author's interpretation. - Gus Wiseman, May 03 2019

			There are a(10) = 27 such partitions of 10:
01:  [ 1 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 1 1 1 3 ]
04:  [ 1 1 1 1 1 1 4 ]
05:  [ 1 1 1 1 1 2 3 ]
06:  [ 1 1 1 1 1 5 ]
07:  [ 1 1 1 1 2 4 ]
08:  [ 1 1 1 1 6 ]
09:  [ 1 1 1 2 5 ]
10:  [ 1 1 1 7 ]
11:  [ 1 1 2 6 ]
12:  [ 1 1 3 5 ]
13:  [ 1 1 8 ]
14:  [ 1 2 3 4 ]
15:  [ 1 2 7 ]
16:  [ 1 3 6 ]
17:  [ 1 9 ]
18:  [ 2 2 2 2 2 ]
19:  [ 2 2 2 4 ]
20:  [ 2 2 6 ]
21:  [ 2 3 5 ]
22:  [ 2 8 ]
23:  [ 3 3 4 ]
24:  [ 3 7 ]
25:  [ 4 6 ]
26:  [ 5 5 ]
27:  [ 10 ]

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500 (terms 0..203 from Joerg Arndt)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A240027 (strictly increasing differences).
Cf. A179255 (distinct parts, nondecreasing), A179254 (distinct parts, strictly increasing).
Cf. A007294, A049988, A320466, A320470, A325325, A325354, A325356, A325360.

Table[Length[Select[IntegerPartitions[n],OrderedQ[Differences[#]]&]],{n,0,30}] (* Gus Wiseman, May 03 2019 *)

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def f(n)
  return 1 if n == 0
  cnt = 0
  partition(n, 1, n).each{|ary|
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    cnt += 1 if ary0.sort == ary0.reverse
  }
  cnt
end
def A240026(n)
  (0..n).map{|i| f(i)}
end
p A240026(50) # Seiichi Manyama, Oct 13 2018

A320466 Number of partitions of n such that the successive differences of consecutive parts are nonincreasing.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 7, 9, 12, 12, 13, 18, 17, 21, 25, 24, 27, 34, 33, 38, 44, 43, 47, 58, 56, 62, 70, 70, 78, 90, 84, 96, 109, 108, 118, 132, 127, 140, 158, 158, 167, 189, 185, 204, 221, 218, 236, 260, 261, 282, 301, 299, 322, 358, 350, 376, 405, 404, 432, 472, 466, 500

Offset: 0

Seiichi Manyama, Oct 13 2018

nonn

Partitions are usually written with parts in descending order, but the conditions are easier to check "visually" if written in ascending order.
Partitions (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) >= p(k) - p(k-1) for all k >= 3.
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). Then a(n) is the number of integer partitions of n whose differences are weakly decreasing. The Heinz numbers of these partitions are given by A325361. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are weakly decreasing, which is the author's interpretation. - Gus Wiseman, May 03 2019

			There are a(10) = 12 such partitions of 10:
01: [10]
02: [1, 9]
03: [2, 8]
04: [3, 7]
05: [4, 6]
06: [5, 5]
07: [1, 4, 5]
08: [2, 4, 4]
09: [1, 2, 3, 4]
10: [1, 3, 3, 3]
11: [2, 2, 2, 2, 2]
12: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
There are a(11) = 13 such partitions of 11:
01: [11]
02: [1, 10]
03: [2, 9]
04: [3, 8]
05: [4, 7]
06: [5, 6]
07: [1, 4, 6]
08: [1, 5, 5]
09: [2, 4, 5]
10: [3, 4, 4]
11: [2, 3, 3, 3]
12: [1, 2, 2, 2, 2, 2]
13: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A240026, A240027, A320470.
Cf. A320382 (distinct parts, nonincreasing).
Cf. A049988, A320509, A325325, A325350, A325353, A325361.

Table[Length[Select[IntegerPartitions[n],GreaterEqual@@Differences[#]&]],{n,0,30}] (* Gus Wiseman, May 03 2019 *)

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def f(n)
  return 1 if n == 0
  cnt = 0
  partition(n, 1, n).each{|ary|
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    cnt += 1 if ary0.sort == ary0
  }
  cnt
end
def A320466(n)
  (0..n).map{|i| f(i)}
end
p A320466(50)

A240027 Number of partitions of n such that the successive differences of consecutive parts are strictly increasing.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 5, 7, 9, 9, 13, 14, 16, 20, 23, 25, 32, 34, 38, 45, 51, 55, 65, 70, 77, 89, 99, 106, 122, 131, 143, 161, 177, 189, 211, 229, 248, 272, 298, 317, 349, 378, 406, 440, 479, 511, 554, 597, 640, 686, 744, 792, 850, 913, 973, 1039, 1122, 1189, 1268, 1358, 1444, 1532, 1646, 1742, 1847, 1975, 2094, 2210, 2366

Offset: 0

Joerg Arndt, Mar 31 2014

nonn

Partitions (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) < p(k) - p(k-1) for all k >= 3.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). Then a(n) is the number of integer partitions of n whose differences are strictly increasing. The Heinz numbers of these partitions are given by A325456. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are strictly increasing, which is the author's interpretation. - Gus Wiseman, May 03 2019

			There are a(15) = 25 such partitions of 15:
01:  [ 1 1 2 4 7 ]
02:  [ 1 1 2 11 ]
03:  [ 1 1 3 10 ]
04:  [ 1 1 4 9 ]
05:  [ 1 1 13 ]
06:  [ 1 2 4 8 ]
07:  [ 1 2 12 ]
08:  [ 1 3 11 ]
09:  [ 1 4 10 ]
10:  [ 1 14 ]
11:  [ 2 2 3 8 ]
12:  [ 2 2 4 7 ]
13:  [ 2 2 11 ]
14:  [ 2 3 10 ]
15:  [ 2 4 9 ]
16:  [ 2 13 ]
17:  [ 3 3 9 ]
18:  [ 3 4 8 ]
19:  [ 3 12 ]
20:  [ 4 4 7 ]
21:  [ 4 11 ]
22:  [ 5 10 ]
23:  [ 6 9 ]
24:  [ 7 8 ]
25:  [ 15 ]

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..800 (terms 0..260 from Joerg Arndt)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A240026 (nondecreasing differences).
Cf. A179255 (distinct parts, nondecreasing), A179254 (distinct parts, strictly increasing).
Cf. A049988, A179269, A320466, A320470, A325325, A325357, A325391, A325456.

Table[Length[Select[IntegerPartitions[n],Less@@Differences[#]&]],{n,0,30}] (* Gus Wiseman, May 03 2019 *)

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def f(n)
  return 1 if n == 0
  cnt = 0
  partition(n, 1, n).each{|ary|
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    cnt += 1 if ary0.sort == ary0.reverse && ary0.uniq == ary0
  }
  cnt
end
def A240027(n)
  (0..n).map{|i| f(i)}
end
p A240027(50) # Seiichi Manyama, Oct 13 2018

A320509 Number of partitions of n such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 6, 4, 6, 8, 7, 8, 11, 7, 12, 14, 10, 13, 19, 12, 18, 21, 16, 19, 27, 19, 25, 30, 25, 30, 37, 25, 35, 40, 35, 42, 49, 35, 49, 56, 46, 54, 66, 50, 65, 72, 60, 70, 83, 68, 84, 90, 80, 94, 110, 86, 107, 116, 98, 119, 137, 111, 134, 146, 130, 148, 165, 141, 169

Offset: 0

Seiichi Manyama, Oct 14 2018

nonn

Partitions are usually written with parts in descending order, but the conditions are easier to check visually if written in ascending order.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences (with the first part taken to be 0) of (6,3,1) are (-3,-2,-1). Then a(n) is the number of integer partitions of n whose differences (with the last part taken to be 0) are weakly decreasing. The Heinz numbers of these partitions are given by A325364. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences (with the first part taken to be 0) are weakly decreasing, which is the author's interpretation. - Gus Wiseman, May 03 2019

			There are a(11) = 8 such partitions of 11:
01: [11]
02: [4, 7]
03: [5, 6]
04: [2, 4, 5]
05: [3, 4, 4]
06: [2, 3, 3, 3]
07: [1, 2, 2, 2, 2, 2]
08: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
There are a(12) = 11 such partitions of 12:
01: [12]
02: [4, 8]
03: [5, 7]
04: [6, 6]
05: [2, 4, 6]
06: [3, 4, 5]
07: [4, 4, 4]
08: [3, 3, 3, 3]
09: [1, 2, 3, 3, 3]
10: [2, 2, 2, 2, 2, 2]
11: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..2000 (terms 0..300 from Seiichi Manyama)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A240026, A240027, A320466, A320470, A320510.
Cf. A320387 (distinct parts, nonincreasing, and first difference <= first part).
Cf. A007294, A007862, A325324, A325350, A325353, A325364, A325390.

Table[Length[Select[IntegerPartitions[n],GreaterEqual@@Differences[Append[#,0]]&]],{n,0,30}] (* Gus Wiseman, May 03 2019 *)

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def f(n)
  return 1 if n == 0
  cnt = 0
  partition(n, 1, n).each{|ary|
    ary << 0
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    cnt += 1 if ary0.sort == ary0
  }
  cnt
end
def A320509(n)
  (0..n).map{|i| f(i)}
end
p A320509(50)

A325548 Number of compositions of n with strictly decreasing differences.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 10, 13, 19, 23, 29, 38, 46, 55, 69, 80, 96, 115, 132, 154, 183, 207, 238, 276, 314, 356, 405, 455, 513, 579, 647, 724, 809, 897, 998, 1107, 1225, 1350, 1486, 1639, 1805, 1973, 2166, 2374, 2586, 2824, 3084, 3346, 3646, 3964, 4286, 4655, 5047

Offset: 0

Gus Wiseman, May 10 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

			The a(1) = 1 through a(8) = 19 compositions:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)    (8)
       (11)  (12)  (13)   (14)   (15)    (16)   (17)
             (21)  (22)   (23)   (24)    (25)   (26)
                   (31)   (32)   (33)    (34)   (35)
                   (121)  (41)   (42)    (43)   (44)
                          (122)  (51)    (52)   (53)
                          (131)  (132)   (61)   (62)
                          (221)  (141)   (133)  (71)
                                 (231)   (142)  (134)
                                 (1221)  (151)  (143)
                                         (232)  (152)
                                         (241)  (161)
                                         (331)  (233)
                                                (242)
                                                (251)
                                                (332)
                                                (341)
                                                (431)
                                                (1331)

Alois P. Heinz, Table of n, a(n) for n = 0..400
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A011782, A000740, A008965, A070211, A175342, A179254, A320470, A325457, A325545, A325546, A325547, A325552, A325557.

b:= proc(n, l, d) option remember; `if`(n=0, 1, add(`if`(l=0 or
       j-l b(n, 0$2):
seq(a(n), n=0..52);  # Alois P. Heinz, Jan 27 2024

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Greater@@Differences[#]&]],{n,0,15}]

a(26)-a(44) from Lars Blomberg, May 30 2019

a(45)-a(52) from Alois P. Heinz, Jan 27 2024

A325457 Heinz numbers of integer partitions with strictly decreasing differences.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98

Offset: 1

Gus Wiseman, May 03 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The enumeration of these partitions by sum is given by A320470.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   6: {1,2}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A056239, A112798, A320470, A320510, A325328, A325352, A325360, A325361, A325368, A325399, A325456, A325461, A320470, A325396.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Greater@@Differences[primeptn[#]]&]

A325399 Heinz numbers of integer partitions whose k-th differences are strictly decreasing for all k >= 0.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 70, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115

Offset: 1

Gus Wiseman, May 02 2019

nonn

First differs from A167171 in having 70. First differs from A325398 in lacking 42.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The enumeration of these partitions by sum is given by A325393.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   31: {11}
   33: {2,5}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

A subsequence of A005117.

Cf. A056239, A112798, A320466, A320470, A325358, A325391, A325393, A325396, A325397, A325398, A325400, A325405, A325457, A325467.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],And@@Table[Greater@@Differences[primeptn[#],k],{k,0,PrimeOmega[#]}]&]

A320510 Number of partitions of n such that the successive differences of consecutive parts are decreasing, and first difference < first part.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 3, 4, 3, 4, 6, 3, 5, 6, 5, 6, 9, 5, 7, 9, 8, 8, 11, 8, 11, 13, 10, 12, 15, 11, 15, 16, 14, 16, 21, 15, 20, 22, 18, 21, 26, 21, 24, 28, 25, 28, 33, 26, 32, 34, 33, 36, 40, 34, 40, 45, 40, 43, 49, 43, 52, 54, 49, 54, 62, 56, 62, 64, 61, 67, 75, 66

Offset: 0

Seiichi Manyama, Oct 14 2018

nonn

Partitions are usually written with parts in descending order, but the conditions are easier to check visually if written in ascending order.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1). Then a(n) is the number of integer partitions of n whose differences (with the last part taken to be 0) are strictly decreasing. The Heinz numbers of these partitions are given by A325461. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are strictly decreasing, which is the author's interpretation. - Gus Wiseman, May 04 2019

			There are a(29) = 13 such partitions of 29:
01: [29]
02: [10, 19]
03: [11, 18]
04: [12, 17]
05: [13, 16]
06: [14, 15]
07: [6, 10, 13]
08: [6, 11, 12]
09: [7, 10, 12]
10: [7, 11, 11]
11: [8, 10, 11]
12: [9, 10, 10]
13: [4, 7, 9, 9]
There are a(30) = 10 such partitions of 30:
01: [30]
02: [11, 19]
03: [12, 18]
04: [13, 17]
05: [14, 16]
06: [15, 15]
07: [6, 11, 13]
08: [7, 11, 12]
09: [8, 11, 11]
10: [4, 7, 9, 10]

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..2000 (terms 0..300 from Seiichi Manyama)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A240026, A240027, A320466, A320470, A320509.
Cf. A320385 (distinct parts, decreasing, and first difference < first part).
Cf. A007294, A007862, A325324, A325358, A325393, A325461.

Table[Length[Select[IntegerPartitions[n],Greater@@Differences[Append[#,0]]&]],{n,0,30}] (* Gus Wiseman, May 04 2019 *)

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def f(n)
  return 1 if n == 0
  cnt = 0
  partition(n, 1, n).each{|ary|
    ary << 0
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    cnt += 1 if ary0.sort == ary0 && ary0.uniq == ary0
  }
  cnt
end
def A320510(n)
  (0..n).map{|i| f(i)}
end
p A320510(50)

A342499 Number of integer partitions of n with strictly decreasing first quotients.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 10, 11, 14, 15, 18, 20, 23, 26, 31, 34, 39, 42, 45, 51, 58, 65, 70, 78, 83, 91, 102, 111, 122, 133, 145, 158, 170, 182, 202, 217, 231, 248, 268, 285, 307, 332, 354, 374, 404, 436, 468, 502, 537, 576, 618, 654, 694, 737, 782, 830

Offset: 0

Gus Wiseman, Mar 17 2021

nonn

Also the number of reversed 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).

			The partition (6,6,3,1) has first quotients (1,1/2,1/3) so is counted under a(16).
The a(1) = 1 through a(9) = 9 partitions:
  (1)  (2)   (3)   (4)   (5)    (6)    (7)    (8)    (9)
       (11)  (21)  (22)  (32)   (33)   (43)   (44)   (54)
                   (31)  (41)   (42)   (52)   (53)   (63)
                         (221)  (51)   (61)   (62)   (72)
                                (321)  (331)  (71)   (81)
                                              (332)  (432)
                                              (431)  (441)
                                                     (531)
                                                     (3321)

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.

The version for differences instead of quotients is A320470.
The ordered version is A342494.
The strictly increasing version is A342498.
The weakly decreasing version is A342513.
The strict case is A342518.
The Heinz numbers of these partitions are listed by A342525.
A000005 counts constant partitions.
A000009 counts strict partitions.
A000041 counts partitions.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A342096 counts partitions with adjacent x < 2y (strict: A342097).
A342098 counts partitions with adjacent parts x > 2y.
Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

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

cp's OEIS Frontend

A325325 Number of integer partitions of n with distinct differences between successive parts.

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A240026 Number of partitions of n such that the successive differences of consecutive parts are nondecreasing.

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A320466 Number of partitions of n such that the successive differences of consecutive parts are nonincreasing.

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A240027 Number of partitions of n such that the successive differences of consecutive parts are strictly increasing.

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A320509 Number of partitions of n such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part.

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A325548 Number of compositions of n with strictly decreasing differences.

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A325457 Heinz numbers of integer partitions with strictly decreasing differences.

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A325399 Heinz numbers of integer partitions whose k-th differences are strictly decreasing for all k >= 0.

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