A304430 -id:A304430 - cp's OEIS frontend

A357864 Numbers whose prime indices have strictly decreasing run-sums. Heinz numbers of the partitions counted by A304430.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 24, 25, 27, 29, 31, 32, 37, 41, 43, 45, 47, 48, 49, 53, 59, 61, 64, 67, 71, 73, 79, 80, 81, 83, 89, 96, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 135, 137, 139, 149, 151, 157, 160, 163, 167, 169, 173

Offset: 1

Gus Wiseman, Oct 19 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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.
The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).

			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}
   23: {9}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   29: {10}
For example, the prime indices of 24 are {1,1,1,2}, with run-sums (3,2), which are strictly decreasing, so 24 is in the sequence.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Subsequence of A304686.
These partitions are counted by A304430.
These are the indices  of rows in A354584 that are strictly decreasing.
The weakly decreasing version is A357861, counted by A304406.
The opposite version is A357862, counted by A304428, complement A357863.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A118914, A181819, A300273, A304405, A304442, A357875.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[300],Greater@@Total/@Split[primeMS[#]]&]

A304442 Number of partitions of n in which the sequence of the sum of the same summands is constant.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 5, 2, 7, 3, 5, 2, 13, 2, 5, 4, 11, 2, 13, 2, 12, 4, 5, 2, 28, 3, 5, 5, 12, 2, 18, 2, 17, 4, 5, 4, 44, 2, 5, 4, 24, 2, 18, 2, 12, 10, 5, 2, 63, 3, 9, 4, 12, 2, 34, 4, 24, 4, 5, 2, 67, 2, 5, 10, 27, 4, 18, 2, 12, 4, 14, 2, 120, 2, 5, 7, 12, 4, 18, 2, 54

Offset: 0

Seiichi Manyama, May 12 2018

nonn

Said differently, these are partitions whose run-sums are all equal. - Gus Wiseman, Jun 25 2022

			a(72) = binomial(d(72),1) + binomial(d(36),2) + binomial(d(24),3) + binomial(d(18),4) + binomial(d(12),6) = 12 + 36 + 56 + 15 + 1 = 120, where d(n) is the number of divisors of n.
--+----------------------+-----------------------------------------
n |                      | Sequence of the sum of the same summands
--+----------------------+-----------------------------------------
1 | 1                    | 1
2 | 2                    | 2
  | 1+1                  | 2
3 | 3                    | 3
  | 1+1+1                | 3
4 | 4                    | 4
  | 2+2                  | 4
  | 2+1+1                | 2, 2
  | 1+1+1+1              | 4
5 | 5                    | 5
  | 1+1+1+1+1            | 5
6 | 6                    | 6
  | 3+3                  | 6
  | 3+1+1+1              | 3, 3
  | 2+2+2                | 6
  | 1+1+1+1+1+1          | 6

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000005 (d(n)), A304405, A304406, A304428, A304430.
All parts are divisors of n, see A018818, compositions A100346.
For run-lengths instead of run-sums we have A047966, compositions A329738.
These partitions are ranked by A353833.
The distinct instead of equal version is A353837, ranked by A353838, compositions A353850.
The version for compositions is A353851, ranked by A353848.
Cf. A098504, A098859, A275870, A353832, A353847, A353864, A353932.

Table[Length[Select[IntegerPartitions[n],SameQ@@Total/@Split[#]&]],{n,0,15}] (* Gus Wiseman, Jun 25 2022 *)

a(n) = if (n==0, 1, sumdiv(n, d, binomial(numdiv(n/d), d))); \\ Michel Marcus, May 13 2018

a(n) >= 2 for n > 1.

a(n) = Sum_{d|n} binomial(A000005(n/d), d) for n > 0.

A304405 Number of partitions of n in which the sequence of the sum of the same summands is nondecreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 12, 18, 22, 31, 37, 52, 61, 80, 97, 127, 147, 189, 220, 277, 325, 402, 469, 578, 665, 804, 933, 1121, 1282, 1537, 1754, 2081, 2374, 2793, 3179, 3739, 4232, 4923, 5587, 6477, 7305, 8445, 9519, 10949, 12323, 14110, 15825, 18099, 20229, 23005

Offset: 0

Seiichi Manyama, May 12 2018

nonn

Number of integer partitions of n with weakly decreasing run-sums, complement A357878. - Gus Wiseman, Oct 22 2022

			n |                      | Sequence of the sum of the same summands
--+----------------------+-----------------------------------------
1 | 1                    | 1
2 | 2                    | 2
  | 1+1                  | 2
3 | 3                    | 3
  | 2+1                  | 1, 2
  | 1+1+1                | 3
4 | 4                    | 4
  | 3+1                  | 1, 3
  | 2+2                  | 4
  | 2+1+1                | 2, 2
  | 1+1+1+1              | 4
5 | 5                    | 5
  | 4+1                  | 1, 4
  | 3+2                  | 2, 3
  | 3+1+1                | 2, 3
  | 2+2+1                | 1, 4
  | 1+1+1+1+1            | 5
6 | 6                    | 6
  | 5+1                  | 1, 5
  | 4+2                  | 2, 4
  | 4+1+1                | 2, 4
  | 3+3                  | 6
  | 3+2+1                | 1, 2, 3
  | 3+1+1+1              | 3, 3
  | 2+2+2                | 6
  | 2+2+1+1              | 2, 4
  | 1+1+1+1+1+1          | 6

The strict opposite version is A304430, ranked by A357864.
The strict version is A304428, ranked by A357862.
The opposite version is A304406, ranked by A357861.
Number of rows in A354584 summing to n that are strictly increasing.
These partitions are ranked by A357875.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with equal run-sums, distinct A353837.
Cf. A047966, A087980, A098859, A100883, A239312, A275870, A353832, A353864.

Table[Length[Select[IntegerPartitions[n],GreaterEqual@@Total/@Split[#]&]],{n,0,30}] (* Gus Wiseman, Oct 22 2022 *)

A304428 Number of partitions of n in which the sequence of the sum of the same summands is increasing.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 11, 14, 20, 26, 33, 41, 50, 64, 81, 97, 120, 150, 176, 210, 255, 303, 362, 426, 503, 595, 703, 816, 953, 1113, 1283, 1482, 1721, 1988, 2299, 2650, 3031, 3464, 3965, 4492, 5115, 5820, 6592, 7467, 8484, 9568, 10822, 12185, 13724, 15445, 17381, 19475, 21855

Offset: 0

Seiichi Manyama, May 12 2018

nonn

Number of integer partitions of n with strictly decreasing run-sums. - Gus Wiseman, Oct 21 2022

			n |                      | Sequence of the sum of the same summands
--+----------------------+-----------------------------------------
1 | 1                    | 1
2 | 2                    | 2
  | 1+1                  | 2
3 | 3                    | 3
  | 2+1                  | 1, 2
  | 1+1+1                | 3
4 | 4                    | 4
  | 3+1                  | 1, 3
  | 2+2                  | 4
  | 1+1+1+1              | 4
5 | 5                    | 5
  | 4+1                  | 1, 4
  | 3+2                  | 2, 3
  | 3+1+1                | 2, 3
  | 2+2+1                | 1, 4
  | 1+1+1+1+1            | 5
6 | 6                    | 6
  | 5+1                  | 1, 5
  | 4+2                  | 2, 4
  | 4+1+1                | 2, 4
  | 3+3                  | 6
  | 3+2+1                | 1, 2, 3
  | 2+2+2                | 6
  | 2+2+1+1              | 2, 4
  | 1+1+1+1+1+1          | 6

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 71 terms from Seiichi Manyama)

The weak version is A304405, ranked by A357875.
The weak opposite version is A304406, ranked by A357861.
The opposite version is A304430, ranked by A357864.
Number of rows in A354584 summing to n that are strictly increasing.
These partitions are ranked by A357862, complement A357863.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with equal run-sums, distinct A353837.
Cf. A047966, A087980, A098859, A100471, A239312, A275870, A353832, A353864.

Table[Length[Select[IntegerPartitions[n],Greater@@Total/@Split[#]&]],{n,0,30}] (* Gus Wiseman, Oct 21 2022 *)

a(n) <= A304405(n).

A304406 Number of partitions of n in which the sequence of the sum of the same summands is nonincreasing.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 6, 5, 9, 8, 11, 11, 20, 16, 20, 21, 32, 30, 41, 38, 50, 48, 62, 64, 89, 81, 97, 100, 123, 123, 151, 154, 187, 183, 221, 221, 279, 272, 312, 316, 377, 376, 446, 460, 531, 547, 628, 641, 754, 746, 841, 856, 990, 1007, 1145, 1167, 1325, 1346, 1519, 1567, 1776

Offset: 0

Seiichi Manyama, May 12 2018

nonn

Number of integer partitions of n with weakly increasing run-sums. - Gus Wiseman, Oct 21 2022

			n |                      | Sequence of the sum of the same summands
--+----------------------+-----------------------------------------
1 | 1                    | 1
2 | 2                    | 2
  | 1+1                  | 2
3 | 3                    | 3
  | 1+1+1                | 3
4 | 4                    | 4
  | 2+2                  | 4
  | 2+1+1                | 2, 2
  | 1+1+1+1              | 4
5 | 5                    | 5
  | 2+1+1+1              | 3, 2
  | 1+1+1+1+1            | 5
6 | 6                    | 6
  | 3+3                  | 6
  | 3+1+1+1              | 3, 3
  | 2+2+2                | 6
  | 2+1+1+1+1            | 4, 2
  | 1+1+1+1+1+1          | 6

Cf. A100882.
These partitions are ranked by A357861.
The complement is A357865, ranked by A357850.
The opposite version is A304405, ranked by A357875.
The strict version is A304430, ranked by A357864.
The strict opposite version is A304428, ranked by A357862.
Number of rows in A354584 summing to n that are weakly decreasing.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with equal run-sums, distinct A353837.
Cf. A047966, A087980, A098859, A239312, A275870, A353832, A353833, A353864.

Table[Length[Select[IntegerPartitions[n],LessEqual@@Total/@Split[#]&]],{n,0,30}] (* Gus Wiseman, Oct 21 2022 *)

A382076 Number of integer partitions of n whose run-sums are not all equal.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 6, 13, 15, 27, 37, 54, 64, 99, 130, 172, 220, 295, 372, 488, 615, 788, 997, 1253, 1547, 1955, 2431, 3005, 3706, 4563, 5586, 6840, 8332, 10139, 12305, 14879, 17933, 21635, 26010, 31181, 37314, 44581, 53156, 63259, 75163, 89124, 105553, 124752, 147210

Offset: 0

Gus Wiseman, Apr 02 2025

nonn

Also the number of integer partitions of n that cannot be partitioned into distinct constant multisets with a common sum. Multiset partitions of this type are ranked by A005117 /\ A326534 /\ A355743, while twice-partitions are counted by A382524, strict case of A279789.

			The partition (3,2,1,1,1) has runs ((3),(2),(1,1,1)) with sums (3,2,3) so is counted under a(8).
The a(3) = 1 through a(8) = 15 partitions:
  (21)  (31)  (32)    (42)     (43)      (53)
              (41)    (51)     (52)      (62)
              (221)   (321)    (61)      (71)
              (311)   (411)    (322)     (332)
              (2111)  (2211)   (331)     (431)
                      (21111)  (421)     (521)
                               (511)     (611)
                               (2221)    (3221)
                               (3211)    (3311)
                               (4111)    (4211)
                               (22111)   (5111)
                               (31111)   (22211)
                               (211111)  (32111)
                                         (311111)
                                         (2111111)

The complement is counted by A304442, ranks A353833.
For distinct instead of equal block-sums we have A381717.
This is the strict case of A381993, see A381995, zeros A381871.
A050361 counts factorizations into distinct prime powers, see A381715.
A304405 counts partitions with weakly decreasing run-sums, ranks A357875.
A304406 counts partitions with weakly increasing run-sums, ranks A357861.
A304428 counts partitions with strictly decreasing run-sums, ranks A357862.
A304430 counts partitions with strictly increasing run-sums, ranks A357864.
A317141 counts coarsenings of prime indices, refinements A300383.
A326534 ranks multiset partitions with a common sum.
A353837 counts partitions with distinct run-sums.
A354584 lists run-sums of weakly increasing prime indices.
A355743 ranks multiset partitions into constant blocks.
Cf. A000688, A005117, A006171, A047966, A279784, A381453, A381455, A381635, A381636, A381994, A382204.

Table[Length[Select[IntegerPartitions[n],!SameQ@@Total/@Split[#]&]],{n,0,15}]

More terms from Bert Dobbelaere, Apr 26 2025

A357861 Numbers whose prime indices have weakly decreasing run-sums. Heinz numbers of the partitions counted by A304406.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 24, 25, 27, 29, 31, 32, 37, 40, 41, 43, 45, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 80, 81, 83, 89, 96, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 135, 137, 139, 144, 149, 151, 157

Offset: 1

Gus Wiseman, Oct 19 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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.
The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).

			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}
   12: {1,1,2}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
For example, the prime indices of 24 are {1,1,1,2}, with run-sums (3,2), which are weakly decreasing, so 24 is in the sequence.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

These partitions are counted by A304406.
These are the indices  of rows in A354584 that are weakly decreasing.
The complement is A357850, counted by A357865, opposite A357876.
The strictly decreasing version is A357864, counted by A304430.
The opposite (weakly increasing) version is A357875, counted by A304405.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A047966, A118914, A181819, A239312, A300273, A304442.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],GreaterEqual@@Total/@Split[primeMS[#]]&]

A383097 Number of integer partitions of n having more than one permutation with all equal run-sums.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 9, 0, 7, 0, 12, 0, 1, 0, 38, 0, 1, 1, 18, 0, 38, 0, 32, 0, 1, 0, 90, 0, 1, 0, 71, 0, 78, 0, 33, 10, 1, 0, 228, 0, 31, 0, 42, 0, 156, 0, 123, 0, 1, 0, 447, 0, 1, 16, 146, 0, 222, 0, 63, 0, 102, 0, 811, 0, 1, 29, 75, 0, 334, 0

Offset: 0

Gus Wiseman, Apr 17 2025

nonn

			The a(27) = 1 partition is: (9,3,3,3,1,1,1,1,1,1,1,1,1).
The a(4) = 1 through a(16) = 9 partitions (empty columns not shown):
  (211)  (3111)  (422)     (511111)  (633)        (71111111)  (844)
                 (41111)             (6222)                   (82222)
                 (221111)            (33222)                  (442222)
                                     (4221111)                (44221111)
                                     (6111111)                (422221111)
                                     (33111111)               (811111111)
                                     (222111111)              (4411111111)
                                                              (42211111111)
                                                              (222211111111)

These partitions are ranked by A383015, positions of terms > 1 in A382877.
For run-lengths instead of sums we have A383090, ranks A383089, unique A383094.
The complement is A383095 + A383096, ranks A383099 \/ A383100.
For any positive number of permutations we have A383098, ranks A383110.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
A382876 counts permutations of prime indices with distinct run-sums, zeros A381636.
Cf. A006171, A329738, A353832, A353839, A353850, A353932, A354584, A381717, A382076.

Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Total/@Split[#]&]]>1&]],{n,0,15}]

More terms from Bert Dobbelaere, Apr 26 2025

A357862 Numbers whose prime indices have strictly increasing run-sums. Heinz numbers of the partitions counted by A304428.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Gus Wiseman, Oct 19 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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.
The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    6: {1,2}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   18: {1,2,2}
   19: {8}
For example, the prime indices of 24 are {1,1,1,2}, with run-sums (3,2), which are not strictly increasing, so 24 is not in the sequence.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

These partitions are counted by A304428.
The complement is A357863.
These are the indices  of rows in A354584 that are strictly increasing.
The opposite (strictly decreasing) version is A357864, counted by A304430.
The weakly increasing version is A357875, counted by A304405.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A118914, A181819, A275870, A300273, A304442.

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

A383095 Number of integer partitions of n having exactly one permutation with all equal run-sums.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 6, 2, 4, 5, 6, 2, 12, 2, 6, 8, 5, 2, 20, 2, 12, 8, 6, 2, 20, 5, 6, 12, 12, 2, 34, 2, 6, 8, 6, 8, 45, 2, 6, 8, 20, 2, 34, 2, 12, 28, 6, 2, 30, 5, 20, 8, 12, 2, 52, 8, 20, 8, 6, 2, 78, 2, 6, 28, 7, 8, 34, 2, 12, 8, 34, 2, 80, 2, 6, 28, 12, 8, 34, 2, 30, 25

Offset: 0

Gus Wiseman, Apr 16 2025

nonn

			The partition (2,2,1,1) has permutation (2,1,1,2) so is counted under a(6).
The a(1) = 1 through a(10) = 6 partitions (A=10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              1111         222              2222      33111      22222
                           2211             11111111  3111111    2221111
                           21111                      111111111  22111111
                           111111                                1111111111

For distinct instead of equal run-sums we have A000005.
For run-lengths instead of sums we have A383094.
The complement is counted by A383096 + A383097, ranks A383100 \/ A383015.
These partitions are ranked by A383099 = positions of 1 in A382877.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
A383098 counts partitions with a permutation having all equal run-sums, ranks A383110.
Cf. A329738, A353839, A353832, A353932, A354584, A381636, A381871, A382076, A382876.

Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], SameQ@@Total/@Split[#]&]]==1&]],{n,0,15}]

More terms from Bert Dobbelaere, Apr 26 2025

cp's OEIS Frontend

A357864 Numbers whose prime indices have strictly decreasing run-sums. Heinz numbers of the partitions counted by A304430.

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A304442 Number of partitions of n in which the sequence of the sum of the same summands is constant.

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A304405 Number of partitions of n in which the sequence of the sum of the same summands is nondecreasing.

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A304428 Number of partitions of n in which the sequence of the sum of the same summands is increasing.

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A304406 Number of partitions of n in which the sequence of the sum of the same summands is nonincreasing.

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A382076 Number of integer partitions of n whose run-sums are not all equal.

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A357861 Numbers whose prime indices have weakly decreasing run-sums. Heinz numbers of the partitions counted by A304406.

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A383097 Number of integer partitions of n having more than one permutation with all equal run-sums.

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