cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A364467 Number of integer partitions of n where some part is the difference of two consecutive parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 9, 13, 21, 28, 42, 55, 78, 106, 144, 187, 255, 325, 429, 554, 717, 906, 1165, 1460, 1853, 2308, 2899, 3582, 4468, 5489, 6779, 8291, 10173, 12363, 15079, 18247, 22124, 26645, 32147, 38555, 46285, 55310, 66093, 78684, 93674, 111104
Offset: 0

Views

Author

Gus Wiseman, Jul 31 2023

Keywords

Comments

In other words, the parts are not disjoint from their own first differences.

Examples

			The a(3) = 1 through a(9) = 13 partitions:
  (21)  (211)  (221)   (42)     (421)     (422)      (63)
               (2111)  (321)    (2221)    (431)      (621)
                       (2211)   (3211)    (521)      (3321)
                       (21111)  (22111)   (3221)     (4221)
                                (211111)  (4211)     (4311)
                                          (22211)    (5211)
                                          (32111)    (22221)
                                          (221111)   (32211)
                                          (2111111)  (42111)
                                                     (222111)
                                                     (321111)
                                                     (2211111)
                                                     (21111111)

Crossrefs

For all differences of pairs parts we have A363225, complement A364345.
The complement is counted by A363260.
For subsets of {1..n} we have A364466, complement A364463.
The strict case is A364536, complement A364464.
These partitions have ranks A364537.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A050291 counts double-free subsets, complement A088808.
A323092 counts double-free partitions, ranks A320340.
A325325 counts partitions with distinct first differences.
Cf. A002865, A025065, A093971, A108917, A196723, A229816, A236912, A237113, A237667, A320347, A326083.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Intersection[#,-Differences[#]]!={}&]],{n,0,30}]
  • Python
    from collections import Counter
from sympy.utilities.iterables import partitions
def A364467(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), partitions(n,size=True)) if not set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023

A386581 Number of normal multisets of size n with no permutation having all distinct run lengths.

Original entry on oeis.org

0, 0, 1, 1, 5, 11, 20, 51, 108, 229, 448, 953, 1940, 3951, 7986, 15972
Offset: 0

Views

Author

Gus Wiseman, Aug 12 2025

Keywords

Comments

A multiset is normal iff it covers an initial interval of positive integers.

Examples

			The normal multiset m = {1,1,1,2,2,2} has permutation (1,2,2,2,1,1) with run lengths (1,3,2), so m is not counted under a(6).
The a(1) = 0 through a(6) = 20 multisets:
  .  (12)  (123)  (1122)  (11123)  (111123)
                  (1123)  (11223)  (111234)
                  (1223)  (11233)  (112233)
                  (1233)  (11234)  (112234)
                  (1234)  (12223)  (112334)
                          (12233)  (112344)
                          (12234)  (112345)
                          (12333)  (122223)
                          (12334)  (122234)
                          (12344)  (122334)
                          (12345)  (122344)
                                   (122345)
                                   (123333)
                                   (123334)
                                   (123344)
                                   (123345)
                                   (123444)
                                   (123445)
                                   (123455)
                                   (123456)

Crossrefs

The complement for partitions appears to be A239455, ranks A351294 or A381432.
For integer partitions we appear to have A351293, ranks A351295 or A381433.
For weakly decreasing multiplicities we appear to have A383710, ranks A382912.
The complement is counted by A386580, see A383708.
A032020 counts normal multisets with distinct multiplicities.
A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).
A098859 counts partitions with distinct multiplicities, compositions A242882.
Cf. A000009, A025065, A047966, A072233, A116540, A130091, A320347, A326083, A382771, A382913, A383706.

Programs

  • Mathematica
    allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
nodrm[y_]:=Select[Permutations[y],UnsameQ@@Length/@Split[#]&];
Table[Length[Select[allnorm[n],nodrm[#]=={}&]],{n,0,7}]

A364536 Number of strict integer partitions of n where some part is a difference of two consecutive parts.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 1, 2, 2, 5, 4, 6, 6, 9, 11, 16, 17, 23, 25, 30, 38, 48, 55, 65, 78, 92, 106, 127, 146, 176, 205, 230, 277, 315, 366, 421, 483, 552, 640, 727, 829, 950, 1083, 1218, 1408, 1577, 1794, 2017, 2298, 2561, 2919, 3255, 3685, 4116, 4638, 5163
Offset: 0

Views

Author

Gus Wiseman, Jul 31 2023

Keywords

Comments

In other words, strict partitions with parts not disjoint from first differences.

Examples

			The a(3) = 1 through a(15) = 11 partitions (A = 10, B = 11, C = 12):
  21  .  .  42   421  431  63   532   542   84    742   743   A5
            321       521  621  541   632   642   841   752   843
                                631   821   651   A21   761   942
                                721   5321  921   5431  842   C21
                                4321        5421  6421  B21   6432
                                            6321  7321  6431  6531
                                                        6521  7431
                                                        7421  7521
                                                        8321  8421
                                                              9321
                                                              54321

Crossrefs

For all differences of pairs we have A363226, non-strict A363225.
For all non-differences of pairs we have A364346, strict A364345.
The strict complement is counted by A364464, non-strict A363260.
For subsets of {1..n} we have A364466, complement A364463.
The non-strict case is A364467, ranks A364537.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A323092 counts double-free partitions, strict A120641.
A325325 counts partitions with distinct first-differences, strict A320347.
Cf. A002865, A025065, A093971, A108917, A229816, A236912, A237113, A237667, A326083, A364347.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,-Differences[#]]!={}&]],{n,0,30}]
  • Python
    from collections import Counter
from sympy.utilities.iterables import partitions
def A364536(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), filter(lambda p:max(p[1].values(),default=1)==1,partitions(n,size=True))) if not set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023

A364537 Heinz numbers of integer partitions where some part is the difference of two consecutive parts.

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, 108, 114, 120, 126, 130, 132, 133, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 195, 198, 204, 210, 216, 222, 228, 231, 234, 240, 246, 252, 258
Offset: 1

Views

Author

Gus Wiseman, Aug 02 2023

Keywords

Comments

In other words, partitions whose parts are not disjoint from their first differences.
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.

Examples

			The partition {3,4,5,7} with Heinz number 6545 has first differences (1,1,2) so is not in the sequence.
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}

Crossrefs

For all differences of pairs the complement is A364347, counted by A364345.
For all differences of pairs we have A364348, counted by A363225.
Subsets of {1..n} of this type are counted by A364466, complement A364463.
These partitions are counted by A364467, complement A363260.
The strict case is A364536, complement A364464.
A050291 counts double-free subsets, complement A088808.
A323092 counts double-free partitions, ranks A320340.
A325325 counts partitions with distinct first differences.
Cf. A002865, A025065, A093971, A108917, A196723, A229816, A236912, A237113, A237667, A320347, A326083.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#],Differences[prix[#]]]!={}&]

A386580 Number of normal multisets of size n having a permutation with all distinct run lengths.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 12, 13, 20, 27, 64, 71, 108, 145, 206, 412
Offset: 0

Views

Author

Gus Wiseman, Aug 07 2025

Keywords

Comments

A multiset is normal iff it covers an initial interval of positive integers.
Conjecture: Also the number of normal multisets of size n having a disjoint family of strict integer partitions, one of each multiplicity.

Examples

			The normal multiset m = {1,1,1,2,2,2} has permutation (1,2,2,2,1,1) with run lengths (1,3,2), so m is counted under a(6).
The a(n) multisets for n = 1..7:
  (1)  (11)  (111)  (1111)  (11111)  (111111)  (1111111)
             (112)  (1112)  (11112)  (111112)  (1111112)
             (122)  (1222)  (11122)  (111122)  (1111122)
                            (11222)  (111222)  (1111222)
                            (12222)  (111223)  (1111223)
                                     (111233)  (1111233)
                                     (112222)  (1112222)
                                     (112223)  (1122222)
                                     (112333)  (1122223)
                                     (122222)  (1123333)
                                     (122233)  (1222222)
                                     (122333)  (1222233)
                                               (1223333)

Crossrefs

For integer partitions we appear to have A239455, ranks A351294 or A381432.
For weakly decreasing multiplicities we appear to have A383708.
The complement is counted by A386581, see A383710 (ranks A382912).
A000041 counts integer partitions, strict A000009.
A032020 counts normal multisets with distinct multiplicities, increasing A000009.
A098859 counts partitions with distinct multiplicities, compositions A242882.
Cf. A025065, A047966, A048767, A072233, A116540, A130091, A320347, A326083, A382771, A382913, A383706.

Programs

  • Mathematica
    allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
nodrm[y_]:=Select[Permutations[y],UnsameQ@@Length/@Split[#]&];
Table[Length[Select[allnorm[n],nodrm[#]!={}&]],{n,0,5}]

A364673 Number of (necessarily strict) integer partitions of n containing all of their own first differences.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 2, 2, 5, 2, 2, 4, 2, 3, 6, 4, 4, 8, 4, 4, 10, 8, 7, 8, 13, 9, 15, 12, 13, 17, 20, 15, 31, 24, 27, 32, 33, 32, 50, 42, 45, 53, 61, 61, 85, 76, 86, 101, 108, 118, 137, 141, 147, 179, 184, 196, 222, 244, 257, 295, 324, 348, 380, 433
Offset: 0

Views

Author

Gus Wiseman, Aug 03 2023

Keywords

Examples

			The partition y = (12,6,3,2,1) has differences (6,3,1,1), and {1,3,6} is a subset of {1,2,3,6,12}, so y is counted under a(24).
The a(n) partitions for n = 1, 3, 6, 12, 15, 18, 21:
  (1)  (3)    (6)      (12)       (15)         (18)         (21)
       (2,1)  (4,2)    (8,4)      (10,5)       (12,6)       (14,7)
              (3,2,1)  (6,4,2)    (8,4,2,1)    (9,6,3)      (12,6,3)
                       (5,4,2,1)  (5,4,3,2,1)  (6,5,4,2,1)  (8,6,4,2,1)
                       (6,3,2,1)               (7,5,3,2,1)  (9,5,4,2,1)
                                               (8,4,3,2,1)  (9,6,3,2,1)
                                                            (10,5,3,2,1)
                                                            (6,5,4,3,2,1)

Links

Crossrefs

Containing all differences: A007862.
Containing no differences: A364464, strict complement A364536.
Containing at least one difference: A364467, complement A363260.
For subsets of {1..n} we have A364671, complement A364672.
A non-strict version is A364674.
For submultisets instead of subsets we have A364675.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A236912 counts sum-free partitions w/o re-used parts, complement A237113.
A325325 counts partitions with distinct first differences.
Cf. A002865, A025065, A196723, A229816, A237667, A320347, A363225, A364272, A364345, A364463, A364537, A370386.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&SubsetQ[#,-Differences[#]]&]],{n,0,30}]
  • Python
    from collections import Counter
def A364673_list(maxn):
    count = Counter()
    for i in range(maxn//3):
        A,f,i = [[(i+1, )]],0,0
        while f == 0:
            A.append([])
            for j in A[i]:
                for k in j:
                    x = j + (j[-1] + k, )
                    y = sum(x)
                    if y <= maxn:
                        A[i+1].append(x)
                        count.update({y})
            if len(A[i+1]) < 1: f += 1
            i += 1
    return [count[z]+1 for z in range(maxn+1)] # John Tyler Rascoe, Mar 09 2024

A342520 Number of strict integer partitions of n with distinct first quotients.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 6, 8, 10, 12, 13, 16, 20, 25, 30, 37, 42, 50, 57, 65, 80, 93, 108, 127, 147, 170, 198, 225, 258, 297, 340, 385, 448, 499, 566, 647, 737, 832, 937, 1064, 1186, 1348, 1522, 1701, 1916, 2157, 2402, 2697, 3013, 3355, 3742, 4190, 4656, 5191
Offset: 0

Views

Author

Gus Wiseman, Mar 20 2021

Keywords

Comments

Also the number of reversed strict integer partitions of n with distinct first quotients.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

Examples

			The strict partition (12,10,5,2,1) has first quotients (5/6,1/2,2/5,1/2) so is not counted under a(30), even though the first differences (-2,-5,-3,-1) are distinct.
The a(1) = 1 through a(13) = 16 partitions (A..D = 10..13):
  1   2   3    4    5    6     7    8     9     A      B      C     D
          21   31   32   42    43   53    54    64     65     75    76
                    41   51    52   62    63    73     74     84    85
                         321   61   71    72    82     83     93    94
                                    431   81    91     92     A2    A3
                                    521   432   532    A1     B1    B2
                                          531   541    542    543   C1
                                          621   631    632    642   643
                                                721    641    651   652
                                                4321   731    732   742
                                                       821    741   751
                                                       5321   831   832
                                                              921   841
                                                                    A21
                                                                    5431
                                                                    7321

Links

Crossrefs

The version for differences instead of quotients is A320347.
The non-strict version is A342514 (ranking: A342521).
The equal instead of distinct version is A342515.
The non-strict ordered version is A342529.
The version for strict divisor chains is A342530.
A000041 counts partitions (strict: A000009).
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A342086 counts strict chains of divisors with strictly increasing quotients.
A342098 counts (strict) partitions with all adjacent parts x > 2y.
Cf. A000005, A003114, A003242, A005117, A018819, A067824, A238710, A253249, A318991, A318992.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Divide@@@Partition[#,2,1]&]],{n,0,30}]

A364675 Number of integer partitions of n whose nonzero first differences are a submultiset of the parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 7, 7, 10, 12, 15, 15, 26, 25, 35, 45, 55, 60, 86, 94, 126, 150, 186, 216, 288, 328, 407, 493, 610, 699, 896, 1030, 1269, 1500, 1816, 2130, 2620, 3029, 3654, 4300, 5165, 5984, 7222, 8368, 9976, 11637, 13771, 15960, 18978, 21896, 25815, 29915
Offset: 0

Views

Author

Gus Wiseman, Aug 04 2023

Keywords

Comments

Conjecture: For subsets of {1..n} instead of partitions of n we have A101925.
Conjecture: The strict version is A154402.

Examples

			The partition y = (3,2,1,1) has first differences (1,1,0), and (1,1) is a submultiset of y, so y is counted under a(7).
The a(1) = 1 through a(8) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (221)    (33)      (421)      (44)
             (111)  (211)   (2111)   (42)      (2221)     (422)
                    (1111)  (11111)  (222)     (3211)     (2222)
                                     (2211)    (22111)    (4211)
                                     (21111)   (211111)   (22211)
                                     (111111)  (1111111)  (32111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)

Crossrefs

For subsets of {1..n} we appear to have A101925, A364671, A364672.
The strict case (no differences of 0) appears to be A154402.
Starting with the distinct parts gives A342337.
For disjoint multisets: A363260, subsets A364463, strict A364464.
For overlapping multisets: A364467, ranks A364537, strict A364536.
For subsets instead of submultisets we have A364673.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A236912 counts sum-free partitions, complement A237113.
A325325 counts partitions with distinct first differences.
Cf. A002865, A007862, A108917, A229816, A237667, A237668, A320347, A363225, A364272, A364345, A364466.

Programs

  • Mathematica
    submultQ[cap_,fat_] := And@@Function[i,Count[fat,i] >= Count[cap,i]] /@ Union[List@@cap];
Table[Length[Select[IntegerPartitions[n], submultQ[Differences[Union[#]],#]&]], {n,0,30}]

A364674 Number of integer partitions of n containing all of their own nonzero first differences.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 7, 11, 13, 17, 18, 32, 30, 44, 54, 70, 78, 114, 125, 171, 205, 257, 302, 408, 464, 592, 711, 892, 1042, 1330, 1543, 1925, 2279, 2787, 3291, 4061, 4727, 5753, 6792, 8197, 9583, 11593, 13505, 16198, 18965, 22548, 26290, 31340, 36363, 43046
Offset: 0

Views

Author

Gus Wiseman, Aug 04 2023

Keywords

Examples

			The partition (10,5,3,3,2,1) has nonzero differences (5,2,1,1) so is counted under a(24).
The a(1) = 1 through a(9) = 13 partitions:
  (1) (2)  (3)   (4)    (5)     (6)      (7)       (8)        (9)
      (11) (21)  (22)   (221)   (33)     (421)     (44)       (63)
           (111) (211)  (2111)  (42)     (2221)    (422)      (333)
                 (1111) (11111) (222)    (3211)    (2222)     (3321)
                                (321)    (22111)   (3221)     (4221)
                                (2211)   (211111)  (4211)     (22221)
                                (21111)  (1111111) (22211)    (32211)
                                (111111)           (32111)    (42111)
                                                   (221111)   (222111)
                                                   (2111111)  (321111)
                                                   (11111111) (2211111)
                                                              (21111111)
                                                              (111111111)

Crossrefs

For no differences we have A363260, subsets A364463, strict A364464.
For at least one difference we have A364467, ranks A364537, strict A364536.
For subsets instead of partitions we have A364671, complement A364672.
The strict case (no differences of 0) is counted by A364673.
For submultisets instead of subsets we have A364675.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A236912 counts sum-free partitions w/o re-used parts, complement A237113.
A325325 counts partitions with distinct first differences.
Cf. A002865, A007862, A025065, A229816, A237667, A320347, A326083, A363225, A364272, A364466.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], SubsetQ[#,Differences[Union[#]]]&]],{n,0,30}]

A364465 Number of subsets of {1..n} with all different first differences of elements.

Original entry on oeis.org

1, 2, 4, 7, 13, 22, 36, 61, 99, 156, 240, 381, 587, 894, 1334, 1967, 2951, 4370, 6406, 9293, 13357, 18976, 27346, 39013, 55437, 78154, 109632, 152415, 210801, 293502, 406664, 561693, 772463, 1058108, 1441796, 1956293, 2639215, 3579542, 4835842, 6523207
Offset: 0

Views

Author

Gus Wiseman, Jul 30 2023

Keywords

Examples

			The a(0) = 1 through a(4) = 13 subsets:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,2}  {4}
                  {1,3}  {1,2}
                  {2,3}  {1,3}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,2,4}
                         {1,3,4}

Links

Crossrefs

For all differences of pairs of elements we have A196723
For partitions instead of subsets we have A325325, strict A320347.
For subset-sums we have A325864, for partitions A108917, A275972.
A007318 counts subsets by length.
A053632 counts subsets by sum.
A363260 counts partitions disjoint from differences, complement A364467.
A364463 counts subsets disjoint from differences, complement A364466.
Cf. A000009, A008289, A011782, A236912, A320348, A325857, A325877, A325878, A326083, A364345, A364346.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],UnsameQ@@Differences[#]&]],{n,0,10}]

Extensions

More terms from Rémy Sigrist, Aug 06 2023
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