A100883 -id:A100883 - cp's OEIS frontend

A317257 Heinz numbers of alternately co-strong integer partitions.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Gus Wiseman, Jul 25 2018

nonn

The first term absent from this sequence but present in A242031 is 180.
A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
    1: {}          16: {1,1,1,1}     32: {1,1,1,1,1}
    2: {1}         17: {7}           33: {2,5}
    3: {2}         19: {8}           34: {1,7}
    4: {1,1}       20: {1,1,3}       35: {3,4}
    5: {3}         21: {2,4}         36: {1,1,2,2}
    6: {1,2}       22: {1,5}         37: {12}
    7: {4}         23: {9}           38: {1,8}
    8: {1,1,1}     24: {1,1,1,2}     39: {2,6}
    9: {2,2}       25: {3,3}         40: {1,1,1,3}
   10: {1,3}       26: {1,6}         41: {13}
   11: {5}         27: {2,2,2}       42: {1,2,4}
   12: {1,1,2}     28: {1,1,4}       43: {14}
   13: {6}         29: {10}          44: {1,1,5}
   14: {1,4}       30: {1,2,3}       45: {2,2,3}
   15: {2,3}       31: {11}          46: {1,9}

Robert Price, Table of n, a(n) for n = 1..10000

Cf. A056239, A100883, A181819, A182850, A242031, A296150, A305732, A317246.
These partitions are counted by A317256.
The complement is A317258.
Totally co-strong partitions are counted by A332275.
Alternately co-strong compositions are counted by A332338.
Alternately co-strong reversed partitions are counted by A332339.
The total version is A335376.
Cf. A182857, A304660, A305563, A316496, A332292, A332340.

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[100],totincQ[Reverse[primeMS[#]]]&]

Updated with corrected terminology by Gus Wiseman, Jun 04 2020

A332579 Number of integer partitions of n covering an initial interval of positive integers with non-unimodal run-lengths.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 7, 8, 10, 14, 19, 22, 30, 36, 43, 56, 69, 80, 101, 121, 141, 172, 202, 234, 282, 332, 384, 452, 527, 602, 706, 815, 929, 1077, 1236, 1403, 1615, 1842, 2082, 2379, 2702, 3044, 3458, 3908, 4388, 4963, 5589, 6252

Offset: 0

Gus Wiseman, Feb 25 2020

nonn

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

Also the number of strict integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.

			The a(10) = 1 through a(16) = 7 partitions:
  33211  332111  3321111  333211    433211     443211      443221
                          33211111  3332111    4332111     3333211
                                    332111111  33321111    4432111
                                               3321111111  33322111
                                                           43321111
                                                           333211111
                                                           33211111111

MathWorld, Unimodal Sequence

The complement is counted by A332577.
Not requiring the partition to cover an initial interval gives A332281.
The opposite version is A332286.
A version for compositions is A332743.
Partitions covering an initial interval of positive integers are A000009.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0-appended first differences are unimodal are A332283.
Compositions whose negated run-lengths are not unimodal are A332727.
Cf. A007052, A100883, A107429, A227038, A332280, A332284, A332638, A332639, A332640, A332671, A332672, A332728.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n],normQ[#]&&!unimodQ[Length/@Split[#]]&]],{n,0,30}]

A332836 Number of compositions of n whose run-lengths are weakly increasing.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 24, 40, 73, 128, 230, 399, 712, 1241, 2192, 3833, 6746, 11792, 20711, 36230, 63532, 111163, 194782, 340859, 596961, 1044748, 1829241, 3201427, 5604504, 9808976, 17170112, 30051470, 52601074, 92063629, 161140256, 282033124, 493637137, 863982135, 1512197655

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

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

Also compositions whose run-lengths are weakly decreasing.

			The a(0) = 1 through a(5) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (121)   (41)
                        (211)   (122)
                        (1111)  (131)
                                (212)
                                (311)
                                (1211)
                                (2111)
                                (11111)
For example, the composition (2,3,2,2,1,1,2,2,2) has run-lengths (1,1,2,2,3) so is counted under a(17).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for the compositions themselves (not run-lengths) is A000041.
The case of partitions is A100883.
The case of unsorted prime signature is A304678, with dual A242031.
Permitting the run-lengths to be weakly decreasing also gives A332835.
The complement is counted by A332871.
Unimodal compositions are A001523.
Compositions that are not unimodal are A115981.
Compositions with equal run-lengths are A329738.
Compositions whose run-lengths are unimodal are A332726.
Cf. A001462, A072704, A072706, A100882, A181819, A329744, A329766, A332641, A332727, A332745, A332833, A332834.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],LessEqual@@Length/@Split[#]&]],{n,0,10}]

step(M, m)={my(n=matsize(M)[1]); for(p=m+1, n, my(v=vector((p-1)\m, i, M[p-i*m,i]), s=vecsum(v)); M[p,]+=vector(#M,i,s-if(i<=#v, v[i]))); M}
seq(n)={my(M=matrix(n+1, n, i, j, i==1)); for(m=1, n, M=step(M, m)); M[1,n]=0; vector(n+1, i, vecsum(M[i,]))/(n-1)} \\ Andrew Howroyd, Dec 31 2020

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

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

A332297 Number of narrowly totally strongly normal integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2

Offset: 0

Gus Wiseman, Feb 15 2020

A partition is narrowly totally strongly normal if either it is empty, a singleton (narrow), or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) that are themselves a narrowly totally strongly normal partition.

			The a(1) = 1, a(2) = 2, a(3) = 3, and a(55) = 4 partitions:
  (1)  (2)    (3)      (55)
       (1,1)  (2,1)    (10,9,8,7,6,5,4,3,2,1)
              (1,1,1)  (5,5,5,5,5,4,4,4,4,3,3,3,2,2,1)
                       (1)^55
For example, starting with the partition (3,3,2,2,1) and repeatedly taking run-lengths gives (3,3,2,2,1) -> (2,2,1) -> (2,1) -> (1,1) -> (2). The first four are normal and have weakly decreasing run-lengths, and the last is a singleton, so (3,3,2,2,1) is counted under a(11).

Normal partitions are A000009.
The non-totally normal version is A316496.
The widely alternating version is A332292.
The non-strong case of compositions is A332296.
The case of compositions is A332336.
The wide version is a(n) - 1 for n > 1.
Cf. A001462, A025487, A100883, A181819, A182850, A317081, A317245, A317256, A317491, A332275, A332277, A332278, A332291, A332337.

tinQ[q_]:=Or[q=={},Length[q]==1,And[Union[q]==Range[Max[q]],GreaterEqual@@Length/@Split[q],tinQ[Length/@Split[q]]]];
Table[Length[Select[IntegerPartitions[n],tinQ]],{n,0,30}]

a(60)-a(80) from Jinyuan Wang, Jun 26 2020

A332337 Number of widely totally strongly normal compositions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 3, 9, 9, 12, 23, 54, 77, 116, 205, 352, 697, 1174, 2013, 3538, 6209, 10830

Offset: 0

Gus Wiseman, Feb 15 2020

A sequence is widely totally strongly normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) that are themselves a widely totally strongly normal sequence.

			The a(1) = 1 through a(8) = 12 compositions:
  (1)  (11)  (12)   (112)   (212)    (123)     (1213)     (1232)
             (21)   (121)   (221)    (132)     (1231)     (2123)
             (111)  (1111)  (11111)  (213)     (1312)     (2132)
                                     (231)     (1321)     (2312)
                                     (312)     (2131)     (2321)
                                     (321)     (3121)     (3212)
                                     (1212)    (11221)    (12131)
                                     (2121)    (12121)    (13121)
                                     (111111)  (1111111)  (21212)
                                                          (22112)
                                                          (111221)
                                                          (11111111)
For example, starting with (22112) and repeated taking run-lengths gives (22112) -> (221) -> (21) -> (11). These are all normal with weakly decreasing run-lengths, and the last is all 1's, so (22112) is counted under a(8).

Normal compositions are A107429.
The case of partitions is A332278.
The non-strong version is A332279.
Heinz numbers in the case of partitions are A332291.
The narrow version is A332336.
The alternating version is A332340.
The co-strong version is this same sequence.
Cf. A025487, A100883, A181819, A317245, A317491, A329744, A332274, A332276, A332277, A332292, A332293, A332296.

totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],LessEqual@@Length/@Split[ptn],totnQ[Length/@Split[ptn]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],totnQ]],{n,0,10}]

For n > 1, a(n) = A332336(n) - 1.

A332727 Number of compositions of n whose run-lengths are not unimodal.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 3, 8, 28, 74, 188, 468, 1120, 2596, 5944, 13324, 29437, 64288, 138929, 297442, 632074, 1333897, 2798352, 5840164, 12132638, 25102232, 51750419, 106346704, 217921161, 445424102, 908376235, 1848753273, 3755839591, 7617835520, 15428584567, 31207263000

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

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

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

			The a(6) = 1 through a(8) = 8 compositions:
  (11211)  (11311)   (11411)
           (111211)  (111311)
           (112111)  (112112)
                     (113111)
                     (211211)
                     (1111211)
                     (1112111)
                     (1121111)

Andrew Howroyd, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Unimodal Sequence.

Looking at the composition itself (not its run-lengths) gives A115981.
The case of partitions is A332281, with complement counted by A332280.
The complement is counted by A332726.
Unimodal compositions are A001523.
Non-unimodal normal sequences are A328509.
Compositions with normal run-lengths are A329766.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0-appended first differences are unimodal are A332283, with complement A332284, with Heinz numbers A332287.
Compositions whose negation is not unimodal are A332669.
Compositions whose run-lengths are weakly increasing are A332836.
Cf. A007052, A072706, A100883, A181819, A227038, A328509, A329744, A329746, A332578, A332638, A332639, A332670, A332741, A332833.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!unimodQ[Length/@Split[#]]&]],{n,0,10}]

a(n) + A332726(n) = 2^(n - 1).

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020

A332746 Number of integer partitions of n such that either the run-lengths or the negated run-lengths are unimodal.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 134, 174, 227, 291, 373, 473, 598, 748, 936, 1163, 1437, 1771, 2170, 2651, 3226, 3916, 4727, 5702, 6846, 8205, 9793, 11681, 13866, 16462, 19452, 22976, 27041, 31820, 37276, 43693, 51023, 59559, 69309, 80664

Offset: 0

Gus Wiseman, Feb 27 2020

nonn

First differs from A000041 at a(14) = 134, A000041(14) = 135.

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

			The only partition not counted under a(14) = 134 is (4,3,3,2,1,1), whose run-lengths (1,2,1,2) are neither unimodal nor is their negation.

MathWorld, Unimodal Sequence

Looking only at the original run-lengths gives A332281.
Looking only at the negated run-lengths gives A332639.
The complement is counted by A332640.
The Heinz numbers of partitions not in this class are A332643.
Unimodal compositions are A001523.
Partitions with unimodal run-lengths are A332280.
Compositions whose negation is unimodal are A332578.
Partitions whose negated run-lengths are unimodal are A332638.
Run-lengths are neither weakly increasing nor weakly decreasing: A332641.
Run-lengths and negated run-lengths are both unimodal: A332745.
Cf. A007052, A025065, A100883, A115981, A181819, A332283, A332577, A332578, A332642, A332669, A332726, A332831.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[IntegerPartitions[n],unimodQ[Length/@Split[#]]||unimodQ[-Length/@Split[#]]&]],{n,0,30}]

A334965 Numbers with strictly increasing prime multiplicities.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 50, 53, 54, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 89, 97, 98, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 147, 149, 151, 157, 162, 163, 167, 169

Offset: 1

Gus Wiseman, May 18 2020

nonn

First differs from A329131 in lacking 150.

Also numbers whose unsorted prime signature is strictly increasing.

			The sequence of terms together with their prime indices begins:
    1: {}            25: {3,3}           64: {1,1,1,1,1,1}
    2: {1}           27: {2,2,2}         67: {19}
    3: {2}           29: {10}            71: {20}
    4: {1,1}         31: {11}            73: {21}
    5: {3}           32: {1,1,1,1,1}     75: {2,3,3}
    7: {4}           37: {12}            79: {22}
    8: {1,1,1}       41: {13}            81: {2,2,2,2}
    9: {2,2}         43: {14}            83: {23}
   11: {5}           47: {15}            89: {24}
   13: {6}           49: {4,4}           97: {25}
   16: {1,1,1,1}     50: {1,3,3}         98: {1,4,4}
   17: {7}           53: {16}           101: {26}
   18: {1,2,2}       54: {1,2,2,2}      103: {27}
   19: {8}           59: {17}           107: {28}
   23: {9}           61: {18}           108: {1,1,2,2,2}

These are the Heinz numbers of the partitions counted by A100471.
Partitions with strictly decreasing run-lengths are A100881.
Partitions with weakly decreasing run-lengths are A100882.
Partitions with weakly increasing run-lengths are A100883.
The weakly decreasing version is A242031.
The weakly increasing version is A304678.
The strictly decreasing version is A304686.
Compositions with strictly increasing or decreasing run-lengths are A333191.
Cf. A000837, A032020, A098859, A100883, A329744, A332835, A333147, A333190.

Select[Range[100],Less@@Last/@FactorInteger[#]&]

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

A317257 Heinz numbers of alternately co-strong integer partitions.

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A332579 Number of integer partitions of n covering an initial interval of positive integers with non-unimodal run-lengths.

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A332836 Number of compositions of n whose run-lengths are weakly increasing.

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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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A332297 Number of narrowly totally strongly normal integer partitions of n.

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A332337 Number of widely totally strongly normal compositions of n.

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A332727 Number of compositions of n whose run-lengths are not unimodal.

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A332746 Number of integer partitions of n such that either the run-lengths or the negated run-lengths are unimodal.

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