A242031 -id:A242031 - cp's OEIS frontend

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

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

A383098 Number of integer partitions of n having at least one permutation with all equal run-sums.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 7, 2, 7, 5, 7, 2, 19, 2, 7, 8, 14, 2, 27, 2, 24, 8, 7, 2, 58, 5, 7, 13, 30, 2, 72, 2, 38, 8, 7, 8, 135, 2, 7, 8, 91, 2, 112, 2, 45, 38, 7, 2, 258, 5, 51, 8, 54, 2, 208, 8, 143, 8, 7, 2, 525, 2, 7, 44, 153, 8, 256, 2, 75, 8, 136, 2, 891, 2, 7, 57, 87, 8

Offset: 0

Gus Wiseman, Apr 17 2025

nonn

			The partition (4,4,4,2,2,1,1,1,1) has permutations (4,2,2,4,1,1,1,1,4) and (4,1,1,1,1,4,2,2,4) so is counted under a(20).
The a(1) = 1 through a(10) = 7 partitions (A=10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              211          222              422       33111      22222
              1111         2211             2222      3111111    511111
                           3111             41111     111111111  2221111
                           21111            221111               22111111
                           111111           11111111             1111111111

For distinct instead of equal run-sums we appear to have A382427.
For run-lengths instead of sums we have A383013, ranked by complement of A382879.
The case of a unique choice is A383095, ranks A383099 = positions of 1 in A382877.
The complement is counted by A383096, ranks A383100 = positions of 0 in A382877.
These partitions are ranked by A383110.
The case of more than one choice is A383097, ranks A383015.
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.
Cf. A006171, A329738, A353832, A353839, A353850, A353932, A354584, A382076, A382857, A382876, A383094, A383112.

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

a(n) = A383097(n) + A383095(n), ranks A383015 \/ A383099.

More terms from Bert Dobbelaere, Apr 26 2025

A383110 Numbers whose prime indices have a permutation with all equal run-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173

Offset: 1

Gus Wiseman, Apr 20 2025

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, sum A056239.

			The prime indices of 144 are {1,1,1,1,2,2}, with permutations with equal run sums (1,1,1,1,2,2), (1,1,2,1,1,2), (2,1,1,2,1,1), (2,2,1,1,1,1), so 144 is in the sequence.
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}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  37: {12}

For distinct run-sums we appear to have complement of A381636 (counted by A381717).
These are the positions of positive terms in A382877.
For run-lengths instead of sums we have complement of A382879, counted by A383013.
For more than one choice we have A383015.
Partitions of this type are counted by A383098.
For a unique choice we have A383099, counted by A383095.
The complement is A383100, counted by A383096.
A056239 adds up prime indices, row sums of A112798.
A304442 counts partitions with equal run-sums, ranks A353833.
A353851 counts compositions with equal run-sums, ranks A353848.
Cf. A047966, A072774, A130091, A242031, A304686, A304678, A334965.
Cf. A351295, A353832, A353837, A353838, A354584, A381871, A382857, A382876, A383097.

Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Total/@Split[#]&]]>0&]

Equals A383015 \/ A383099, counted by A353851 \/ A383095.

A332643 Neither the unsorted prime signature of a(n) nor the negated unsorted prime signature of a(n) is unimodal.

Original entry on oeis.org

2100, 3300, 3900, 4200, 4410, 5100, 5700, 6468, 6600, 6900, 7644, 7800, 8400, 8700, 9300, 9996, 10200, 10500, 10780, 10890, 11100, 11172, 11400, 12300, 12740, 12900, 12936, 13200, 13230, 13524, 13800, 14100, 15210, 15246, 15288, 15600, 15900, 16500, 16660

Offset: 1

Gus Wiseman, Feb 28 2020

nonn

A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The sequence of terms together with their prime indices begins:
   2100: {1,1,2,3,3,4}
   3300: {1,1,2,3,3,5}
   3900: {1,1,2,3,3,6}
   4200: {1,1,1,2,3,3,4}
   4410: {1,2,2,3,4,4}
   5100: {1,1,2,3,3,7}
   5700: {1,1,2,3,3,8}
   6468: {1,1,2,4,4,5}
   6600: {1,1,1,2,3,3,5}
   6900: {1,1,2,3,3,9}
   7644: {1,1,2,4,4,6}
   7800: {1,1,1,2,3,3,6}
   8400: {1,1,1,1,2,3,3,4}
   8700: {1,1,2,3,3,10}
   9300: {1,1,2,3,3,11}
   9996: {1,1,2,4,4,7}
  10200: {1,1,1,2,3,3,7}
  10500: {1,1,2,3,3,3,4}
  10780: {1,1,3,4,4,5}
  10890: {1,2,2,3,5,5}

MathWorld, Unimodal Sequence

Not requiring non-unimodal negation gives A332282.
These are the Heinz numbers of the partitions counted by A332640.
Not requiring non-unimodality gives A332642.
The case of compositions is A332870.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Unsorted prime signature is A124010.
Non-unimodal normal sequences are A328509.
Partitions whose 0-appended first differences are unimodal are A332283, with Heinz numbers the complement of A332287.
Compositions whose negation is unimodal are A332578.
Compositions whose negation is not unimodal are A332669.
Partitions whose 0-appended first differences are not unimodal are A332744, with Heinz numbers A332832.
Numbers whose signature is neither increasing nor decreasing are A332831.
Cf. A007052, A056239, A072704, A112798, A242031, A242414, A332280, A332281, A332288, A332294, A332639, A332728, A332742.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Select[Range[10000],!unimodQ[Last/@FactorInteger[#]]&&!unimodQ[-Last/@FactorInteger[#]]&]

Intersection of A332282 and A332642.

A383096 Number of integer partitions of n having no permutation with all equal run-sums.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 4, 13, 15, 25, 35, 54, 58, 99, 128, 168, 217, 295, 358, 488, 603, 784, 995, 1253, 1517, 1953, 2429, 2997, 3688, 4563, 5532, 6840, 8311, 10135, 12303, 14875, 17842, 21635, 26008, 31177, 37247, 44581, 53062, 63259, 75130, 89096, 105551, 124752, 147015, 173520

Offset: 0

Gus Wiseman, Apr 17 2025

nonn

			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)         (331)     (431)
                             (421)     (521)
                             (511)     (611)
                             (2221)    (3221)
                             (3211)    (3311)
                             (4111)    (4211)
                             (22111)   (5111)
                             (31111)   (22211)
                             (211111)  (32111)
                                       (311111)
                                       (2111111)

For distinct instead of equal run-sums we appear to have A381717, q.v.
For run-lengths instead of sums we have A382915, ranks A382879, by signature A382914.
For more than one permutation we have A383097, ranks A383015.
The complement is counted by A383098, ranks A383110
These partitions are ranked by A383100, positions of 0 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.
A382876 counts permutations of prime indices with distinct run-sums, zeros A381636.
A383095 counts partitions having a unique permutation with equal run-sums, ranks A383099.
Cf. A006171, A329738, A353832, A353839, A353932, A354584, A382076, A382857, A383094.

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

More terms from Bert Dobbelaere, Apr 26 2025

A111059 a(n) = Product_{k=1..n} A005117(k), the product of the first n squarefree positive integers.

Original entry on oeis.org

1, 2, 6, 30, 180, 1260, 12600, 138600, 1801800, 25225200, 378378000, 6432426000, 122216094000, 2566537974000, 56463835428000, 1298668214844000, 33765373585944000, 979195833992376000, 29375875019771280000

Offset: 1

Leroy Quet, Oct 07 2005

nonn

Do all terms belong to A242031 (weakly decreasing prime signature)? - Gus Wiseman, May 14 2021

			Since the first 6 squarefree positive integers are 1, 2, 3, 5, 6, 7, the 6th term of the sequence is 1*2*3*5*6*7 = 1260.
From _Gus Wiseman_, May 14 2021: (Start)
The sequence of terms together with their prime signatures begins:
             1: ()
             2: (1)
             6: (1,1)
            30: (1,1,1)
           180: (2,2,1)
          1260: (2,2,1,1)
         12600: (3,2,2,1)
        138600: (3,2,2,1,1)
       1801800: (3,2,2,1,1,1)
      25225200: (4,2,2,2,1,1)
     378378000: (4,3,3,2,1,1)
    6432426000: (4,3,3,2,1,1,1)
  122216094000: (4,3,3,2,1,1,1,1)
(End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..417

A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A072047 applies Omega to each squarefree number.
A246867 groups squarefree numbers by Heinz weight (row sums: A147655).
A261144 groups squarefree numbers by smoothness (row sums: A054640).
A319246 gives the sum of prime indices of each squarefree number.
A329631 lists prime indices of squarefree numbers (reversed: A319247).
Cf. A001221, A002110, A014466, A070826, A338901, A339360.

Rest[FoldList[Times,1,Select[Range[40],SquareFreeQ]]] (* Harvey P. Dale, Jun 14 2011 *)

m=30;k=1;for(n=1,m,if(issquarefree(n),print1(k=k*n,",")))

More terms from Klaus Brockhaus, Oct 08 2005

A316529 Heinz numbers of totally strong integer partitions.

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, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83

Offset: 1

Gus Wiseman, Jul 29 2018

nonn

First differs from A304678 at a(115) = 151, A304678(115) = 150.
The alternating version first differs from this sequence in having 150 and lacking 450.
An integer partition is totally strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and are themselves a totally strong partition.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			Starting with (3,3,2,1), which has Heinz number 150, and repeatedly taking run-lengths gives (3,3,2,1) -> (2,1,1) -> (1,2), so 150 is not in the sequence.
Starting with (3,3,2,2,1), which has Heinz number 450, and repeatedly taking run-lengths gives (3,3,2,2,1) -> (2,2,1) -> (2,1) -> (1,1) -> (2) -> (1), so 450 is in the sequence.

Cf. A056239, A181819, A182850, A242031, A296150, A305732, A317246.
The enumeration of these partitions by sum is A316496.
The complement is A316597.
The widely normal version is A332291.
The dual version is A335376.
Partitions with weakly decreasing run-lengths are A100882.
Cf. A025487, A100883, A133808, A317257, A332275, A332293, A332297.

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

Updated with corrected terminology by Gus Wiseman, Mar 08 2020

A317092 Positive integers whose prime multiplicities are weakly decreasing and span an initial interval of positive integers.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 91, 92

Offset: 1

Gus Wiseman, Jul 21 2018

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..25000

Cf. A001222, A007916, A055932, A056239, A112798, A124010, A133808, A242031.

Cf. A317087, A317088, A317089, A317090, A317091.

normalQ[m_]:=Union[m]==Range[Max[m]];
Select[Range[2,100],And[normalQ[FactorInteger[#][[All,2]]],OrderedQ[Reverse[FactorInteger[#][[All,2]]]]]&]

is(n) = my (f=factor(n), w=#f~); if (w==0 || f[w,2]!=1, return (0), for (k=1, w-1, if (f[k,2]!=f[k+1,2] && f[k,2]!=1+f[k+1,2], return (0))); return (1)) \\ Rémy Sigrist, Sep 05 2018

A317258 Heinz numbers of integer partitions that are not totally nonincreasing.

Original entry on oeis.org

18, 50, 54, 75, 90, 98, 108, 126, 147, 150, 162, 180, 198, 234, 242, 245, 250, 252, 270, 294, 300, 306, 324, 338, 342, 350, 363, 375, 378, 396, 414, 450, 468, 486, 490, 500, 507, 522, 525, 540, 550, 558, 578, 588, 594, 600, 605, 612, 630, 648, 650, 666, 684

Offset: 1

Gus Wiseman, Jul 25 2018

nonn

An integer partition is totally nonincreasing if either it is empty or a singleton or its multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are weakly decreasing and are themselves a totally nonincreasing integer partition.

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

			Sequence of all integer partitions that are not totally nonincreasing begins: (221), (331), (2221), (332), (3221), (441), (22211), (4221), (442), (3321), (22221), (32211), (5221), (6221), (551), (443), (3331), (42211), (32221), (4421), (33211), (7221), (222211), (661), (8221), (4331), (552), (3332), (42221), (52211), (9221), (33221).

Cf. A056239, A071365, A100883, A112769, A181819, A182850, A242031, A296150, A305733, A317256, A317257.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
totincQ[q_]:=Or[Length[q]<=1,And[OrderedQ[Length/@Split[q]],totincQ[Reverse[Length/@Split[q]]]]];
Select[Range[1000],!totincQ[Reverse[primeMS[#]]]&]

A316597 Heinz numbers of integer partitions that are not totally nondecreasing.

Original entry on oeis.org

12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 150, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208

Offset: 1

Gus Wiseman, Jul 29 2018

nonn

The first term of this sequence that is absent from A112769 is 150.

An integer partition is totally nondecreasing if either it is empty or a singleton or its multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are weakly increasing and, taken in reverse order, are themselves a totally nondecreasing integer partition.

			150 is the Heinz number of (3,3,2,1), with multiplicities (1,1,2), which has multiplicities (2,1), which are decreasing, so 150 does not belong to the sequence.

Complement of A316529.

Cf. A056239, A071365, A112769, A181819, A182850, A242031, A296150, A305733, A316496, A317258.

cp's OEIS Frontend

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

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A383098 Number of integer partitions of n having at least one permutation with all equal run-sums.

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A383110 Numbers whose prime indices have a permutation with all equal run-sums.

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A332643 Neither the unsorted prime signature of a(n) nor the negated unsorted prime signature of a(n) is unimodal.

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A383096 Number of integer partitions of n having no permutation with all equal run-sums.

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A111059 a(n) = Product_{k=1..n} A005117(k), the product of the first n squarefree positive integers.

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A316529 Heinz numbers of totally strong integer partitions.

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A317092 Positive integers whose prime multiplicities are weakly decreasing and span an initial interval of positive integers.

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