A325349 -id:A325349 - cp's OEIS frontend

A383531 Heinz numbers of integer partitions that do not have distinct multiplicities (Wilf) or distinct nonzero 0-appended differences (conjugate Wilf).

6, 21, 30, 36, 42, 60, 65, 66, 70, 78, 84, 90, 102, 105, 110, 114, 120, 126, 132, 133, 138, 140, 150, 154, 156, 165, 168, 174, 180, 186, 198, 204, 210, 216, 220, 222, 228, 231, 234, 238, 240, 246, 252, 258, 264, 270, 273, 276, 280, 282, 286, 294, 300, 306, 308

Offset: 1

Gus Wiseman, May 15 2025

nonn

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}
   21: {2,4}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   60: {1,1,2,3}
   65: {3,6}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
  102: {1,2,7}
  105: {2,3,4}
  110: {1,3,5}
  114: {1,2,8}
  120: {1,1,1,2,3}

These partitions are counted by A383530.
Negating both sides gives A383532, counted by A383507.
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.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A122111 represents conjugation in terms of Heinz numbers.
A325324 counts integer partitions with distinct 0-appended differences, ranks A325367.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A383709 counts Wilf partitions with distinct 0-appended differences, ranks A383712.
Cf. A001223, A047966, A181819, A238745, A320348, A325325, A325349, A325366, A325368, A325388, A381431, A383506.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y, Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],!UnsameQ@@Length/@Split[prix[#]] && !UnsameQ@@Length/@Split[conj[prix[#]]]&]

Equals A130092 /\ A383513.

A325358 Number of integer partitions of n whose augmented differences are strictly decreasing.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 6, 7, 9, 10, 11, 13, 14, 15, 18, 20, 21, 24, 26, 28, 33, 36, 38, 43, 46, 49, 56, 60, 63, 71, 76, 80, 90, 96, 100, 112, 120, 125, 139, 149, 155, 171, 183, 190, 208, 223, 232, 252, 269, 280, 304, 325, 338, 364, 387, 403

Offset: 0

Gus Wiseman, May 01 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 A325396.

			The a(1) = 1 through a(11) = 6 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (10)   (11)
            (21)  (31)  (41)  (42)  (52)   (62)   (63)   (73)   (83)
                              (51)  (61)   (71)   (72)   (82)   (92)
                                    (421)  (521)  (81)   (91)   (101)
                                                  (621)  (631)  (731)
                                                         (721)  (821)

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. A049988, A240026, A320466, A325349, A325350, A325351, A325356, A325357, A325359, A325393.

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

A325406 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 2, 2, 0, 0, 1, 1, 3, 2, 0, 0, 1, 4, 2, 3, 1, 0, 0, 1, 1, 5, 5, 2, 1, 0, 0, 1, 3, 5, 6, 3, 3, 1, 0, 0, 1, 3, 4, 8, 7, 1, 4, 2, 0, 0, 1, 3, 6, 11, 7, 5, 2, 4, 2, 1, 0, 1, 1, 6, 13, 8, 9, 9, 0, 4, 3, 1, 0, 1, 6, 7, 11, 12, 9

Offset: 0

Gus Wiseman, May 03 2019

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 distinct differences of any degree are the union of the k-th differences for all k >= 0. For example, the k-th differences of (1,1,2,4) for k = 0...3 are:
  (1,1,2,4)
  (0,1,2)
  (1,1)
  (0)
so there are a total of 4 distinct differences of any degree, namely {0,1,2,4}.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  2  0
  0  1  2  2  0
  0  1  1  3  2  0
  0  1  4  2  3  1  0
  0  1  1  5  5  2  1  0
  0  1  3  5  6  3  3  1  0
  0  1  3  4  8  7  1  4  2  0
  0  1  3  6 11  7  5  2  4  2  1
  0  1  1  6 13  8  9  9  0  4  3  1
  0  1  6  7 11 12  9 10  8  4  3  2  2
  0  1  1  7 18  9 14 19  5 10  3  5  4  1
  0  1  3  9 17  9 22 20 15  9  7  6  5  4  1
  0  1  4  8 22 11 16 24 22 19 10 11  2  8  7  2
  0  1  4 10 23 15 24 23 27 27 12 14 11  8  8  5  5
Row n = 8 counts the following partitions:
  (8)  (44)        (17)       (116)     (134)   (1133)   (111122)
       (2222)      (26)       (125)     (233)   (11123)
       (11111111)  (35)       (1115)    (1223)  (11222)
                   (224)      (1124)
                   (1111112)  (11114)
                              (111113)

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

Row sums are A000041.

Cf. A049597, A049988, A279945, A320348, A325324, A325325, A325349, A325404, A325466.

Table[Length[Select[Reverse/@IntegerPartitions[n],Length[Union@@Table[Differences[#,i],{i,0,Length[#]}]]==k&]],{n,0,16},{k,0,n}]

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

A383532 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct nonzero 0-appended differences (conjugate Wilf).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 49, 50, 52, 53, 56, 59, 61, 64, 67, 68, 71, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 97, 98, 99, 101, 103, 104, 107, 109, 112, 113, 116, 117, 121, 124, 125

Offset: 1

Gus Wiseman, May 15 2025

nonn

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: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}

Partitions of this type are counted by A383507.
Negating both sides gives A383531, counted by A383530.
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.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A122111 represents conjugation in terms of Heinz numbers.
A325324 counts integer partitions with distinct 0-appended differences, ranks A325367.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A383709 counts Wilf partitions with distinct 0-appended differences, ranks A383712.
Cf. A001223, A047966, A181819, A238745, A320348, A325325, A325349, A325366, A325368, A325388, A383506.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
paug[y_]:=-DeleteCases[Differences[Append[y,0]],0];
Select[Range[100], UnsameQ@@Last/@FactorInteger[#] && UnsameQ@@paug[Reverse[prix[#]]]&]

Equals A130091 /\ A383512.

A383534 Irregular triangle read by rows where row n lists the positive first differences of the 0-prepended prime indices of n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 2, 5, 1, 1, 6, 1, 3, 2, 1, 1, 7, 1, 1, 8, 1, 2, 2, 2, 1, 4, 9, 1, 1, 3, 1, 5, 2, 1, 3, 10, 1, 1, 1, 11, 1, 2, 3, 1, 6, 3, 1, 1, 1, 12, 1, 7, 2, 4, 1, 2, 13, 1, 1, 2, 14, 1, 4, 2, 1, 1, 8, 15, 1, 1, 4, 1, 2, 2, 5, 1, 5, 16, 1, 1, 3, 2

Offset: 1

Gus Wiseman, May 20 2025

Also differences of distinct 0-prepended prime indices of n.

			The prime indices of 140 are {1,1,3,4}, zero prepended {0,1,1,3,4}, differences (1,0,2,1), positive (1,2,1).
Rows begin:
    1: ()        16: (1)        31: (11)
    2: (1)       17: (7)        32: (1)
    3: (2)       18: (1,1)      33: (2,3)
    4: (1)       19: (8)        34: (1,6)
    5: (3)       20: (1,2)      35: (3,1)
    6: (1,1)     21: (2,2)      36: (1,1)
    7: (4)       22: (1,4)      37: (12)
    8: (1)       23: (9)        38: (1,7)
    9: (2)       24: (1,1)      39: (2,4)
   10: (1,2)     25: (3)        40: (1,2)
   11: (5)       26: (1,5)      41: (13)
   12: (1,1)     27: (2)        42: (1,1,2)
   13: (6)       28: (1,3)      43: (14)
   14: (1,3)     29: (10)       44: (1,4)
   15: (2,1)     30: (1,1,1)    45: (2,1)

Row-lengths are A001221, sums A061395.
Rows start with A055396, end with A241919.
For multiplicities instead of differences we have A124010 (prime signature).
Including difference 0 gives A287352, without prepending A355536.
Positions of first appearances of rows are A358137.
Positions of strict rows are A383512, counted by A098859.
Positions of non-strict rows are A383513, counted by A336866.
Heinz numbers of rows are A383535.
Restricting to rows of squarefree index gives A384008.
Without prepending we get A384009.
A000040 lists the primes, differences A001223.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A320348 counts strict partitions with distinct 0-appended differences, ranks A325388.
A325324 counts partitions with distinct 0-appended differences, ranks A325367.
Cf. A005117, A122111, A130091, A325325, A325349, A325366, A325368, A381431, A383506.

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

a(A005117(n)) = A384008(n).

A325553 Number of compositions of n with distinct circular differences up to sign.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 7, 21, 31, 41, 87, 99, 191, 245, 381, 501, 735, 883, 1309, 1841, 2589, 3435, 4941, 6857, 9791, 13503, 19475, 27073, 37175, 52299, 72249, 100359, 139317, 190549, 256769, 355193, 471963, 644433, 858793, 1159161, 1530879, 2056073, 2711921

Offset: 0

Gus Wiseman, May 11 2019

nonn

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

The circular differences of a composition c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2).

			The a(1) = 1 through a(8) = 21 compositions:
  (1)  (2)  (3)  (4)  (5)  (6)  (7)    (8)
                                (124)  (125)
                                (142)  (134)
                                (214)  (143)
                                (241)  (152)
                                (412)  (215)
                                (421)  (251)
                                       (314)
                                       (341)
                                       (413)
                                       (431)
                                       (512)
                                       (521)
                                       (1124)
                                       (1142)
                                       (1241)
                                       (1421)
                                       (2114)
                                       (2411)
                                       (4112)
                                       (4211)

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

Cf. A000079, A008965, A167606, A173258, A325324, A325349, A325545, A325549, A325551, A325552, A325553, A325556, A325558.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Abs[Differences[Append[#,First[#]]]]&]],{n,20}]

a(0) and a(26)-a(43) from Alois P. Heinz, Jan 28 2024

A383507 Number of Wilf and conjugate Wilf integer partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 7, 9, 12, 14, 19, 20, 27, 30, 31, 40, 50, 56, 68, 76, 86, 112, 126, 139, 170, 197, 216, 251, 297, 317, 378, 411, 466, 521, 607, 621, 745, 791, 892, 975, 1123, 1163, 1366, 1439, 1635, 1757, 2021, 2080, 2464, 2599, 2882, 3116, 3572, 3713

Offset: 0

Gus Wiseman, May 14 2025

nonn

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 a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (1111)  (11111)  (222)     (331)      (332)
                                     (411)     (511)      (611)
                                     (3111)    (4111)     (2222)
                                     (111111)  (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)

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

A048768 gives Look-and-Say fixed points, counted by A217605.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A239455 counts Look-and-Say partitions, complement A351293.
A325349 counts partitions with distinct augmented differences, ranks A325366.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A381431 is the section-sum transform, union A381432, complement A381433.
A383534 gives 0-prepended differences by rank, see A325351.
A383709 counts Wilf partitions with distinct 0-appended differences.
Cf. A047966, A048767, A111133, A320348, A325324, A325325, A325367, A325368, A325388, A351294, A383506.

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

These partitions have Heinz numbers A130091 /\ A383512.

A383712 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct 0-appended differences.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 20, 23, 25, 28, 29, 31, 37, 41, 43, 44, 45, 47, 49, 50, 52, 53, 59, 61, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 113, 116, 117, 121, 124, 127, 131, 137, 139, 148, 149, 151, 153, 157, 163, 164

Offset: 1

Gus Wiseman, May 15 2025

nonn

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.

Integer partitions with distinct multiplicities are called Wilf partitions.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   28: {1,1,4}
   29: {10}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   44: {1,1,5}
   45: {2,2,3}
   47: {15}
   49: {4,4}
   50: {1,3,3}

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

For just distinct multiplicities we have A130091 (conjugate A383512), counted by A098859.
For just distinct 0-appended differences we have A325367, counted by A325324.
These partitions are counted by A383709.
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.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A383507 counts partitions that are Wilf and conjugate Wilf, ranks A383532.
A383530 counts partitions that are not Wilf or conjugate-Wilf, ranks A383531.
Cf. A000720, A005117, A047966, A238745, A320348, A325325, A325349, A325355, A325366, A325368, A325388, A383506.

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

Equals A130091 /\ A325367.

A325466 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree > 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 3, 2, 1, 0, 0, 1, 5, 4, 0, 1, 0, 0, 1, 4, 6, 3, 0, 1, 0, 0, 1, 6, 6, 4, 3, 1, 1, 0, 0, 1, 6, 10, 4, 2, 4, 1, 2, 0, 0, 1, 7, 12, 8, 3, 3, 4, 1, 2, 1, 0, 1, 6, 13, 11, 2, 11, 3, 4, 0, 3, 1, 1, 1, 10, 16, 7, 10, 10

Offset: 0

Gus Wiseman, May 04 2019

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.

			Triangle begins:
  1
  1  0
  1  1  0
  1  2  0  0
  1  3  1  0  0
  1  3  2  1  0  0
  1  5  4  0  1  0  0
  1  4  6  3  0  1  0  0
  1  6  6  4  3  1  1  0  0
  1  6 10  4  2  4  1  2  0  0
  1  7 12  8  3  3  4  1  2  1  0
  1  6 13 11  2 11  3  4  0  3  1  1
  1 10 16  7 10 10  6  6  5  1  1  2  1
  1  7 18 14  7 16 11  6  4  8  0  5  0  1
  1  9 20 18 10 20 13 10 10  4  5  5  2  2  2
  1 10 26 18 10 24 13 19 13 10  6  6  2  8  1  2
  1 11 25 24 16 28 19 24 14 15  9 10  9  5  2  7  1
Row 7 counts the following reversed partitions (empty columns not shown):
  (7)  (16)       (115)     (133)   (11122)
       (25)       (124)     (1123)
       (34)       (223)     (1222)
       (1111111)  (1114)
                  (11113)
                  (111112)
Row 9 counts the following reversed partitions (empty columns not shown):
(9)  (18)         (117)       (126)    (1125)   (1134)    (11223)  (111222)
     (27)         (135)       (144)    (11124)  (1224)             (1111122)
     (36)         (225)       (1233)            (11133)
     (45)         (234)       (12222)           (111123)
     (333)        (1116)
     (111111111)  (2223)
                  (11115)
                  (111114)
                  (1111113)
                  (11111112)

Row sums are A000041. Column k = 1 is A088922.

Cf. A098859, A279945, A325242, A325324, A325325, A325349, A325404, A325405, A325406, A325468.

Table[Length[Select[Reverse/@IntegerPartitions[n],Length[Union@@Table[Differences[#,i],{i,1,Length[#]}]]==k&]],{n,0,16},{k,0,n}]

cp's OEIS Frontend

A383531 Heinz numbers of integer partitions that do not have distinct multiplicities (Wilf) or distinct nonzero 0-appended differences (conjugate Wilf).

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A325358 Number of integer partitions of n whose augmented differences are strictly decreasing.

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A325406 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree.

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

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A383532 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct nonzero 0-appended differences (conjugate Wilf).

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A383534 Irregular triangle read by rows where row n lists the positive first differences of the 0-prepended prime indices of n.

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A325553 Number of compositions of n with distinct circular differences up to sign.

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A383507 Number of Wilf and conjugate Wilf integer partitions of n.

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