A325353 -id:A325353 - cp's OEIS frontend

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

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)

A325404 Number of reversed integer partitions y of n such that the k-th differences of y are distinct for all k >= 0 and are disjoint from the i-th differences for i != k.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 4, 4, 5, 7, 5, 11, 12, 11, 12, 20, 15, 24, 22, 27, 28, 37, 28, 45, 43, 48, 50, 66, 58, 79, 72, 84, 87, 112, 106, 135, 128, 158, 147, 186, 180, 218, 220, 265, 246, 304, 303, 354, 340, 412, 418, 471, 463, 538, 543, 642, 600, 711, 755

Offset: 0

Gus Wiseman, May 02 2019

nonn

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 Heinz numbers of these partitions are given by A325405.

			The a(1) = 1 through a(12) = 5 reversed partitions (A = 10, B = 11, C = 12):
  (1)  (2)  (3)  (4)   (5)   (6)   (7)   (8)   (9)   (A)   (B)    (C)
                 (13)  (14)  (15)  (16)  (17)  (18)  (19)  (29)   (39)
                       (23)        (25)  (26)  (27)  (28)  (38)   (57)
                                   (34)  (35)  (45)  (37)  (47)   (1B)
                                                     (46)  (56)   (2A)
                                                           (1A)
                                                           (146)

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

Cf. A279945, A325325, A325349, A325353, A325354, A325365, A325368, A325391, A325393, A325405, A325406, A325468.

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

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)

A325468 Number of integer partitions y of n such that the k-th differences of y are distinct (independently) for all k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 10, 15, 17, 19, 24, 31, 26, 40, 43, 51, 52, 72, 66, 89, 88, 111, 119, 150, 130, 183, 193, 229, 231, 279, 287, 358, 365, 430, 426, 538, 535, 649, 680, 742, 803, 943, 982, 1136, 1115

Offset: 0

Gus Wiseman, May 03 2019

nonn

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 Heinz numbers of these partitions are given by A325467.

			The a(1) = 1 through a(9) = 6 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)
            (21)  (31)  (32)  (42)  (43)   (53)   (54)
                        (41)  (51)  (52)   (62)   (63)
                                    (61)   (71)   (72)
                                    (421)  (431)  (81)
                                           (521)  (621)

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

Cf. A000009, A325324, A325325, A325349, A325353, A325354, A325391, A325393, A325404, A325406, A325467.

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

A325354 Number of reversed integer partitions of n whose k-th differences are weakly increasing for all k.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 11, 15, 19, 24, 25, 36, 37, 43, 54, 63, 64, 80, 81, 100, 113, 122, 123, 151, 166, 178, 195, 217, 218, 269, 270, 295, 316, 332, 372, 424, 425, 447, 472, 547, 550, 616, 617, 659, 750, 777, 782, 862, 885, 995, 1032, 1083, 1090, 1176, 1275

Offset: 0

Gus Wiseman, May 02 2019

nonn

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 Heinz numbers of these partitions are given by A325400.

			The a(1) = 1 through a(8) = 15 reversed partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (111)  (22)    (23)     (24)      (25)       (26)
                    (112)   (113)    (33)      (34)       (35)
                    (1111)  (1112)   (114)     (115)      (44)
                            (11111)  (123)     (124)      (116)
                                     (222)     (223)      (125)
                                     (1113)    (1114)     (224)
                                     (11112)   (11113)    (1115)
                                     (111111)  (111112)   (1124)
                                               (1111111)  (2222)
                                                          (11114)
                                                          (111113)
                                                          (1111112)
                                                          (11111111)

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

Cf. A007294, A240026, A325353, A325356, A325360, A325362, A325391, A325393, A325394, A325400, A325404, A325406, A325468.

Table[Length[Select[Sort/@IntegerPartitions[n],And@@Table[OrderedQ[Differences[#,k]],{k,0,Length[#]}]&]],{n,0,30}]

A325391 Number of reversed integer partitions of n whose k-th differences are strictly increasing for all k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 5, 6, 8, 9, 9, 13, 13, 15, 19, 20, 20, 28, 28, 30, 36, 40, 40, 50, 50, 56, 64, 68, 68, 86, 86, 92, 102, 112, 114, 133, 133, 146, 158, 173, 173, 202, 202, 215, 237, 256, 256, 287, 287, 324, 340, 359, 359, 403, 423, 446, 464, 495, 495

Offset: 0

Gus Wiseman, May 02 2019

nonn

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 Heinz numbers of these partitions are given by A325398.

			The a(1) = 1 through a(9) = 6 reversed partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)
            (12)  (13)  (14)  (15)  (16)   (17)   (18)
                        (23)  (24)  (25)   (26)   (27)
                                    (34)   (35)   (36)
                                    (124)  (125)  (45)
                                                  (126)
The smallest reversed strict partition with strictly increasing differences not counted by this sequence is (1,2,4,7), whose first and second differences are (1,2,3) and (1,1) respectively.

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

Cf. A179254, A240026, A325353, A325354, A325357, A325393, A325395, A325398, A325404, A325406, A325456, A325468.

Table[Length[Select[Reverse/@IntegerPartitions[n],And@@Table[Less@@Differences[#,k],{k,0,Length[#]}]&]],{n,0,30}]

A325393 Number of integer partitions of n whose k-th differences are strictly decreasing for all k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 8, 7, 9, 11, 10, 12, 15, 13, 16, 19, 18, 20, 24, 22, 26, 29, 28, 31, 37, 33, 38, 43, 42, 44, 52, 48, 55, 59, 58, 62, 72, 65, 74, 80, 80, 82, 94, 88, 99, 103, 104, 108, 123, 114, 126, 133, 135, 137, 155, 145, 161, 166, 169, 174

Offset: 0

Gus Wiseman, May 02 2019

nonn

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 Heinz numbers of these partitions are given by A325399.

			The a(1) = 1 through a(9) = 5 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)   (8)    (9)
            (21)  (31)  (32)  (42)  (43)  (53)   (54)
                        (41)  (51)  (52)  (62)   (63)
                                    (61)  (71)   (72)
                                          (431)  (81)

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

Cf. A049988, A320466, A325353, A325354, A325358, A325391, A325396, A325399, A325404, A325406, A325457, A325468.

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

A325350 Number of integer partitions of n whose augmented differences are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 10, 13, 17, 21, 26, 32, 38, 46, 56, 66, 78, 92, 106, 124, 145, 166, 191, 220, 249, 284, 325, 366, 413, 468, 523, 586, 659, 733, 817, 913, 1011, 1121, 1245, 1373, 1515, 1674, 1838, 2020, 2223, 2433, 2664, 2920, 3184, 3476, 3797, 4129, 4492

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The Heinz numbers of these partitions are given by A325389.

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (31)    (32)     (42)      (52)       (53)
             (111)  (211)   (41)     (51)      (61)       (62)
                    (1111)  (311)    (321)     (421)      (71)
                            (2111)   (411)     (511)      (521)
                            (11111)  (3111)    (3211)     (611)
                                     (21111)   (4111)     (4211)
                                     (111111)  (31111)    (5111)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
For example, (4,2,1,1) has augmented differences (3,2,1,1), which are weakly decreasing, so (4,2,1,1) is counted under a(8).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500

Cf. A007294, A098859, A240026, A320466, A320509, A325349, A325353, A325354, A325356, A325357, A325358, A325361, A325364.

Mathematica
```
aug[y_]:=Table[If[i
				
```

G.f.: Sum_{k>=0} x^k / Product_{j=1..k} (1 - x^(j*(j+1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 25 2019

A325397 Heinz numbers of integer partitions whose k-th differences are weakly decreasing for all k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 87, 89

Offset: 1

Gus Wiseman, May 02 2019

nonn

First differs from A325361 in lacking 150.
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 A325353.

			Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   72: {1,1,1,2,2}
   76: {1,1,8}
   78: {1,2,6}
   80: {1,1,1,1,3}
The first partition that has weakly decreasing differences (A320466, A325361) but is not represented in this sequence is (3,3,2,1), which has Heinz number 150 and whose first and second differences are (0,-1,-1) and (-1,0) respectively.

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

Cf. A056239, A112798, A320466, A320509, A325353, A325361, A325364, A325389, A325398, A325399, A325400, A325405, A325467.

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

cp's OEIS Frontend

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

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A325404 Number of reversed integer partitions y of n such that the k-th differences of y are distinct for all k >= 0 and are disjoint from the i-th differences for i != k.

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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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A325468 Number of integer partitions y of n such that the k-th differences of y are distinct (independently) for all k >= 0.

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A325354 Number of reversed integer partitions of n whose k-th differences are weakly increasing for all k.

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A325391 Number of reversed integer partitions of n whose k-th differences are strictly increasing for all k >= 0.

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A325393 Number of integer partitions of n whose k-th differences are strictly decreasing for all k >= 0.

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A325350 Number of integer partitions of n whose augmented differences are weakly decreasing.

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