A325391 -id:A325391 - cp's OEIS frontend

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

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

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}]

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}]

A179269 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are increasing, and first difference > first part.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7, 7, 10, 10, 10, 13, 14, 14, 18, 19, 19, 23, 25, 25, 30, 32, 33, 38, 41, 42, 48, 52, 54, 60, 65, 67, 75, 81, 84, 92, 99, 103, 113, 121, 126, 136, 147, 153, 165, 177, 184, 197, 213, 221, 236, 253, 264, 280, 301, 313, 331, 355, 371, 390, 418, 435, 458

Offset: 0

Joerg Arndt, Jan 05 2011

nonn

Conditions as in A179254; additionally, if more than 1 part, first difference > first part.

Equivalently, number of partitions for which the sequence of part counts by decreasing part size is 1, 2, 3, ... - Olivier Gérard, Jul 28 2017

			a(10) = 5 as there are 5 such partitions of 10: 1 + 3 + 6 = 1 + 9 = 2 + 8 = 3 + 7 = 10.
a(10) = 5 as there are 5 such partitions of 10: 10, 8 + 1 + 1, 6 + 2 + 2, 4 + 3 + 3, 3 + 2 + 2 + 1 + 1 + 1 (second definition).
From _Gus Wiseman_, May 04 2019: (Start)
The a(3) = 1 through a(13) = 7 partitions whose differences are strictly increasing (with the last part taken to be 0) are the following (A = 10, B = 11, C = 12, D = 13). The Heinz numbers of these partitions are given by A325460.
  (3)  (4)   (5)   (6)   (7)   (8)   (9)   (A)    (B)    (C)    (D)
       (31)  (41)  (51)  (52)  (62)  (72)  (73)   (83)   (93)   (94)
                         (61)  (71)  (81)  (82)   (92)   (A2)   (A3)
                                           (91)   (A1)   (B1)   (B2)
                                           (631)  (731)  (831)  (C1)
                                                                (841)
                                                                (931)
The a(3) = 1 through a(11) = 5 partitions whose multiplicities form an initial interval of positive integers are the following (A = 10, B = 11). The Heinz numbers of these partitions are given by A307895.
  (3)  (4)    (5)    (6)    (7)    (8)    (9)    (A)       (B)
       (211)  (311)  (411)  (322)  (422)  (522)  (433)     (533)
                            (511)  (611)  (711)  (622)     (722)
                                                 (811)     (911)
                                                 (322111)  (422111)
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..10000 (terms 0..200 from Seiichi Manyama)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A179254 (condition only on differences), A007294 (nondecreasing instead of strictly increasing), A179255, A320382, A320385, A320387, A320388.

Cf. A007862, A240027, A307895, A320509, A320510, A325324, A325357, A325391, A325460.

Table[Length@
  Select[IntegerPartitions[n],
   And @@ Equal[Range[Length[Split[#]]], Length /@ Split[#]] &], {n,
0, 40}]   (* Olivier Gérard, Jul 28 2017 *)
Table[Length[Select[IntegerPartitions[n],Less@@Differences[Append[#,0]]&]],{n,0,30}] (* Gus Wiseman, May 04 2019 *)

R(n)={my(L=List(), v=vectorv(n, i, 1), w=1, t=1); while(v, listput(L,v); w++; t+=w; v=vectorv(n, i, sum(k=1, (i-1)\t, L[w-1][i-k*t]))); Mat(L)}
seq(n)={my(M=R(n)); concat([1], vector(n, i, vecsum(M[i,])))} \\ Andrew Howroyd, Aug 27 2019

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).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.reverse && ary0.uniq == ary0
  }
  cnt
end
def A179269(n)
  (0..n).map{|i| f(i)}
end
p A179269(50) # Seiichi Manyama, Oct 12 2018

def A179269(n):
    has_increasing_diffs = lambda x: min(differences(x,2)) >= 1
    special = lambda x: (x[1]-x[0]) > x[0]
    allowed = lambda x: (len(x) < 2 or special(x)) and (len(x) < 3 or has_increasing_diffs(x))
    return len([x for x in Partitions(n,max_slope=-1) if allowed(x[::-1])])
# D. S. McNeil, Jan 06 2011

G.f.: Sum_{k>=0} x^(k*(k+1)*(k+2)/6) / Product_{j=1..k} (1 - x^(j*(j+1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 25 2019

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}]

A325357 Number of integer partitions of n whose augmented differences are strictly increasing.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 3, 5, 5, 4, 5, 6, 5, 7, 7, 7, 7, 9, 7, 10, 10, 8, 11, 13, 10, 13, 14, 12, 14, 17, 13, 17, 19, 17, 18, 22, 19, 22, 24, 21, 24, 28, 24, 29, 30, 28, 31, 35, 30, 35, 40, 36

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 A325395.

			The a(28) = 10 partitions:
  (28)
  (18,10)
  (17,11)
  (16,12)
  (15,13)
  (14,14)
  (12,10,6)
  (11,10,7)
  (10,10,8)
  (8,8,7,5)
For example, the augmented differences of (8,8,7,5) are (1,2,3,5), which are strictly increasing.

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..2000

Cf. A000837, A007294, A049988, A098859, A325351, A325356, A325360, A325391, A325395.

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

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}]

A325398 Heinz numbers of reversed integer partitions whose k-th differences are strictly increasing 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, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109

Offset: 1

Gus Wiseman, May 02 2019

nonn

First differs from A301899 in lacking 105. First differs from A325399 in having 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 A325391.

			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, A325357, A325391, A325395, A325397, A325399, A325400, A325405, A325406, A325456, A325467.

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

A325353 Number of integer partitions of n whose k-th differences are weakly decreasing for all k >= 0.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 7, 9, 11, 12, 13, 17, 16, 19, 23, 23, 24, 30, 29, 35, 37, 37, 40, 49, 47, 51, 56, 59, 61, 73, 65, 75, 80, 84, 91, 99, 91, 103, 112, 120, 114, 132, 126, 143, 154, 147, 152, 175, 169, 190, 187, 194, 198, 226, 225, 231, 236, 246, 256, 293

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 A325397.

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (321)     (2221)     (332)
                                     (111111)  (1111111)  (431)
                                                          (2222)
                                                          (11111111)
The first partition that has weakly decreasing differences (A320466) but is not counted under a(9) is (3,3,2,1), 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. A320466, A320509, A325350, A325354, A325391, A325393, A325397, A325398, A325399, A325404, A325405, A325406, A325468.

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

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[#]}]&]

cp's OEIS Frontend

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

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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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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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A179269 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are increasing, and first difference > first part.

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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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A325357 Number of integer partitions of n whose augmented differences are strictly increasing.

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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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A325398 Heinz numbers of reversed integer partitions whose k-th differences are strictly increasing for all k >= 0.

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