A325367 -id:A325367 - cp's OEIS frontend

A320348 Number of partition into distinct parts (a_1, a_2, ... , a_m) (a_1 > a_2 > ... > a_m and Sum_{k=1..m} a_k = n) such that a1 - a2, a2 - a_3, ... , a_{m-1} - a_m, a_m are different.

1, 1, 1, 2, 3, 2, 4, 4, 4, 6, 9, 7, 13, 12, 13, 16, 22, 17, 28, 28, 31, 36, 50, 45, 63, 62, 74, 78, 102, 92, 123, 123, 146, 148, 191, 181, 228, 233, 280, 283, 348, 350, 420, 437, 518, 523, 616, 641, 727, 774, 884, 911, 1038, 1102, 1240, 1292, 1463, 1530, 1715, 1861, 2002

Offset: 1

Seiichi Manyama, Oct 11 2018

nonn

Also the number of integer partitions of n whose parts cover an initial interval of positive integers with distinct multiplicities. Also the number of integer partitions of n whose multiplicities cover an initial interval of positive integers and are distinct (see A048767 for a bijection). - Gus Wiseman, May 04 2019

			n = 9
[9]        *********  a_1 = 9.
           ooooooooo
------------------------------------
[8, 1]             *        a_2 = 1.
            *******o  a_1 - a_2 = 7.
            oooooooo
------------------------------------
[7, 2]            **        a_2 = 2.
             *****oo  a_1 - a_2 = 5.
             ooooooo
------------------------------------
[5, 4]          ****        a_2 = 4.
               *oooo  a_1 - a_2 = 1.
               ooooo
------------------------------------
a(9) = 4.
From _Gus Wiseman_, May 04 2019: (Start)
The a(1) = 1 through a(11) = 9 strict partitions with distinct differences (where the last part is taken to be 0) are the following (A = 10, B = 11). The Heinz numbers of these partitions are given by A325388.
  (1)  (2)  (3)  (4)   (5)   (6)   (7)   (8)   (9)   (A)    (B)
                 (31)  (32)  (51)  (43)  (53)  (54)  (64)   (65)
                       (41)        (52)  (62)  (72)  (73)   (74)
                                   (61)  (71)  (81)  (82)   (83)
                                                     (91)   (92)
                                                     (631)  (A1)
                                                            (632)
                                                            (641)
                                                            (731)
The a(1) = 1 through a(10) = 6 partitions covering an initial interval of positive integers with distinct multiplicities are the following. The Heinz numbers of these partitions are given by A325326.
  1  11  111  211   221    21111   2221     22211     22221      222211
              1111  2111   111111  22111    221111    2211111    322111
                    11111          211111   2111111   21111111   2221111
                                   1111111  11111111  111111111  22111111
                                                                 211111111
                                                                 1111111111
The a(1) = 1 through a(10) = 6 partitions whose multiplicities cover an initial interval of positive integers and are distinct are the following (A = 10). The Heinz numbers of these partitions are given by A325337.
  (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)    (9)    (A)
                 (211)  (221)  (411)  (322)  (332)  (441)  (433)
                        (311)         (331)  (422)  (522)  (442)
                                      (511)  (611)  (711)  (622)
                                                           (811)
                                                           (322111)
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..500 (terms 1..100 from Seiichi Manyama)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000009, A320347.

Cf. A007294, A007862, A048767, A098859, A179269, A320509, A320510, A325324, A325325, A325349, A325367, A325404, A325468.

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

A325324 Number of integer partitions of n whose differences (with the last part taken to be 0) are distinct.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 4, 7, 7, 7, 10, 15, 13, 22, 25, 26, 31, 43, 39, 55, 54, 68, 75, 98, 97, 128, 135, 165, 177, 217, 223, 277, 282, 339, 356, 438, 444, 527, 553, 667, 694, 816, 868, 1015, 1054, 1279, 1304, 1538, 1631, 1849, 1958, 2304, 2360, 2701, 2899, 3267

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

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).

The Heinz numbers of these partitions are given by A325367.

			The a(1) = 1 through a(11) = 15 partitions (A = 10, B = 11):
  (1)  (2)   (3)  (4)   (5)    (6)    (7)    (8)    (9)    (A)    (B)
       (11)       (22)  (32)   (33)   (43)   (44)   (54)   (55)   (65)
                  (31)  (41)   (51)   (52)   (53)   (72)   (64)   (74)
                        (311)  (411)  (61)   (62)   (81)   (73)   (83)
                                      (322)  (71)   (441)  (82)   (92)
                                      (331)  (332)  (522)  (91)   (A1)
                                      (511)  (611)  (711)  (433)  (443)
                                                           (622)  (533)
                                                           (631)  (551)
                                                           (811)  (632)
                                                                  (641)
                                                                  (722)
                                                                  (731)
                                                                  (911)
                                                                  (6311)
For example, (6,3,1,1) has differences (-3,-2,0,-1), which are distinct, so (6,3,1,1) is counted under a(11).

Fausto A. C. Cariboni, 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. A007862, A049988, A098859, A130091, A240026, A320348, A320466, A320509, A325325, A325349, A325366, A325367, A325368, A325404, A325407.

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

A325368 Heinz numbers of integer partitions with distinct differences between successive parts.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A325325.

			Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
    8: {1,1,1}
   16: {1,1,1,1}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}
   72: {1,1,1,2,2}
   80: {1,1,1,1,3}
   81: {2,2,2,2}
   88: {1,1,1,5}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}
  100: {1,1,3,3}

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

Cf. A056239, A112798, A130091, A240026, A325325, A325328, A325352, A325360, A325361, A325366, A325367, A325405, A325456, A325457.

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

A325366 Heinz numbers of integer partitions whose augmented differences are distinct.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 99, 101, 102, 103

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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 enumeration of these partitions by sum is given by A325349.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   25: {3,3}
   26: {1,6}
   29: {10}
   31: {11}

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

Positions of squarefree numbers in A325351.

Cf. A056239, A093641, A112798, A130091, A325349, A325355, A325367, A325368, A325389, A325394, A325395, A325396, A325405.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
aug[y_]:=Table[If[i

A383512 Heinz numbers of conjugate Wilf partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85

Offset: 1

Gus Wiseman, May 13 2025

nonn

First differs from A364347 in having 130 and lacking 110.
First differs from A381432 in lacking 65 and 133.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

			The terms together with their prime indices begin:
     1: {}           17: {7}            35: {3,4}
     2: {1}          19: {8}            37: {12}
     3: {2}          20: {1,1,3}        38: {1,8}
     4: {1,1}        22: {1,5}          39: {2,6}
     5: {3}          23: {9}            40: {1,1,1,3}
     7: {4}          25: {3,3}          41: {13}
     8: {1,1,1}      26: {1,6}          43: {14}
     9: {2,2}        27: {2,2,2}        44: {1,1,5}
    10: {1,3}        28: {1,1,4}        45: {2,2,3}
    11: {5}          29: {10}           46: {1,9}
    13: {6}          31: {11}           47: {15}
    14: {1,4}        32: {1,1,1,1,1}    49: {4,4}
    15: {2,3}        33: {2,5}          50: {1,3,3}
    16: {1,1,1,1}    34: {1,7}          51: {2,7}

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

Partitions of this type are counted by A098859.
The conjugate version is A130091, complement A130092.
Including differences of 0 gives A325367, counted by A325324.
The strict case is A325388, counted by A320348.
The complement is A383513, counted by A336866.
Also requiring distinct multiplicities gives A383532, counted by A383507.
These are the positions of strict rows in A383534, or squarefree numbers in A383535.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A325349 counts partitions with distinct augmented differences, ranks A325366.
A383530 counts partitions that are not Wilf or conjugate Wilf, ranks A383531.
A383709 counts Wilf partitions with distinct augmented differences, ranks A383712.
Cf. A000720, A005117, A048768, A109298, A325325, A325351, A325355, A325368, A381431, A383506.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], UnsameQ@@DeleteCases[Differences[Prepend[prix[#],0]],0]&]

A383513 Heinz numbers of non conjugate Wilf partitions.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 70, 72, 78, 84, 90, 96, 102, 105, 108, 110, 114, 120, 126, 132, 133, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 198, 204, 210, 216, 220, 222, 228, 231, 234, 238, 240, 246

Offset: 1

Gus Wiseman, May 13 2025

nonn

First differs from A381433 in having 65.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

			The terms together with their prime indices begin:
    6: {1,2}
   12: {1,1,2}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   60: {1,1,2,3}
   63: {2,2,4}
   65: {3,6}
   66: {1,2,5}
   70: {1,3,4}
   72: {1,1,1,2,2}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}

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

Partitions of this type are counted by A336866.
The conjugate version is A130092, complement A130091.
Including differences of 0 gives complement of A325367, counted by A325324.
The strict case is the complement of A325388, counted by A320348.
The complement is A383512, counted by A098859.
Also forbidding distinct multiplicities gives A383531, counted by A383530.
These are positions of non-strict rows in A383534, or nonsquarefree numbers in A383535.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A383507 counts partitions that are Wilf and conjugate Wilf, ranks A383532.
A383709 counts Wilf partitions with distinct augmented differences, ranks A383712.
Cf. A000720, A005117, A048768, A238745, A325325, A325351, A325355, A325366, A325368, A381431, A383506.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!UnsameQ@@DeleteCases[Differences[Prepend[prix[#],0]],0]&]

A325362 Heinz numbers of integer partitions whose differences (with the last part taken to be 0) are weakly increasing.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 78, 79, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 110, 111, 113

Offset: 1

Gus Wiseman, May 02 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 (x, y, z) are (y - x, z - y). We adhere to this standard for integer partitions also even though they are always weakly decreasing. For example, the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).
The enumeration of these partitions by sum is given by A007294.
This sequence and A025487, considered as sets, are related by the partition conjugation function A122111(.), which maps the members of either set 1:1 onto the other set. - Peter Munn, Feb 10 2022

			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}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   30: {1,2,3}
   31: {11}
   33: {2,5}

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

Cf. A007294, A056239, A112798, A240026, A320348, A325327, A325360, A325364, A325367, A325390, A325394, A325400.

Related to A025487 via A122111.

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

A325388 Heinz numbers of strict integer partitions with distinct differences (with the last part taken to be 0).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 122

Offset: 1

Gus Wiseman, May 02 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) (with the last part taken to be 0) are (-3,-2,-1).
The enumeration of these partitions by sum is given by A320348.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   19: {8}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   31: {11}
   33: {2,5}
   34: {1,7}
   35: {3,4}

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

A subsequence of A005117.

Cf. A056239, A112798, A320348, A325324, A325327, A325362, A325364, A325366, A325367, A325368, A325390, A325405, A325460, A325461, A325467.

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

A325405 Heinz numbers of integer partitions y 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, 2, 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 122

Offset: 1

Gus Wiseman, May 02 2019

nonn

First differs from A325388 in lacking 130.
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 A325404.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   19: {8}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   31: {11}
   33: {2,5}
   34: {1,7}
   35: {3,4}

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

A subsequence of A005117.

Cf. A056239, A112798, A279945, A325325, A325366, A325367, A325368, A325397, A325398, A325399, A325400, A325404, A325406, A325467.

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

A325390 Heinz number of the negated differences plus one of the integer partition with Heinz number n (with the last part taken to be 0).

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 10, 15, 13, 18, 17, 21, 15, 24, 19, 18, 23, 30, 25, 33, 29, 36, 14, 39, 20, 42, 31, 27, 37, 48, 35, 51, 21, 36, 41, 57, 55, 60, 43, 45, 47, 66, 30, 69, 53, 72, 22, 30, 65, 78, 59, 36, 35, 84, 85, 87, 61, 54, 67, 93, 50, 96, 49, 63, 71

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of a positive integer sequence (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) (with the last part taken to be 0) are (-3,-2,-1).

			The Heinz number of (6,3,1) is 130, and its negated differences plus one are (4,3,2), which has Heinz number 105, so a(130) = 105.

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

Number of appearances of n is A325392(n).
Positions of squarefree numbers are A325367.
Cf. A007294, A007862, A056239, A112798, A320509, A325324, A325327, A325351, A325352, A325362, A325364, A325460, A325461.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Times@@Prime/@(1-Differences[Append[primeptn[n],0]]),{n,100}]

cp's OEIS Frontend

A320348 Number of partition into distinct parts (a_1, a_2, ... , a_m) (a_1 > a_2 > ... > a_m and Sum_{k=1..m} a_k = n) such that a1 - a2, a2 - a_3, ... , a_{m-1} - a_m, a_m are different.

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A325324 Number of integer partitions of n whose differences (with the last part taken to be 0) are distinct.

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A325368 Heinz numbers of integer partitions with distinct differences between successive parts.

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A325366 Heinz numbers of integer partitions whose augmented differences are distinct.

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A383512 Heinz numbers of conjugate Wilf partitions.

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A383513 Heinz numbers of non conjugate Wilf partitions.

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A325362 Heinz numbers of integer partitions whose differences (with the last part taken to be 0) are weakly increasing.

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A325388 Heinz numbers of strict integer partitions with distinct differences (with the last part taken to be 0).

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