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.

Previous Showing 11-19 of 19 results.

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[#]]]!={}&]

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

A364671 Number of subsets of {1..n} containing all of their own first differences.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 23, 34, 58, 96, 171, 302, 565, 1041, 1969, 3719, 7105, 13544, 25999, 49852, 95949, 184658, 356129, 687068, 1327540, 2566295, 4966449, 9617306, 18640098, 36150918, 70166056, 136272548, 264844111, 515036040, 1002211421, 1951345157, 3801569113
Offset: 0

Views

Author

Gus Wiseman, Aug 04 2023

Keywords

Examples

			The subset {1,2,4,5,10,14} has differences (1,2,1,5,4) so is counted under a(14).
The a(0) = 1 through a(5) = 14 subsets:
  {}  {}   {}     {}       {}         {}
      {1}  {1}    {1}      {1}        {1}
           {2}    {2}      {2}        {2}
           {1,2}  {3}      {3}        {3}
                  {1,2}    {4}        {4}
                  {1,2,3}  {1,2}      {5}
                           {2,4}      {1,2}
                           {1,2,3}    {2,4}
                           {1,2,4}    {1,2,3}
                           {1,2,3,4}  {1,2,4}
                                      {1,2,3,4}
                                      {1,2,3,5}
                                      {1,2,4,5}
                                      {1,2,3,4,5}

Links

Crossrefs

For differences of all strict pairs we have A054519, for partitions A007862.
For "disjoint" instead of "subset" we have A364463, partitions A363260.
For "non-disjoint" we have A364466, partitions A364467 (strict A364536).
The complement is counted by A364672, partitions A364673, A364674, A364675.
First differences of terms are A364752, complement A364753.
Cf. A151897, A196723, A237667, A237668, A325325, A326083, A363225, A364345, A364464, A364537.

Programs

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

Extensions

More terms from Rémy Sigrist, Aug 06 2023

A364672 Number of subsets of {1..n} not containing all of their own first differences.

Original entry on oeis.org

0, 0, 0, 2, 6, 18, 41, 94, 198, 416, 853, 1746, 3531, 7151, 14415, 29049, 58431, 117528, 236145, 474436, 952627, 1912494, 3838175, 7701540, 15449676, 30988137, 62142415, 124600422, 249795358, 500719994, 1003575768, 2011211100, 4030123185, 8074898552, 16177657763, 32408393211, 64917907623
Offset: 0

Views

Author

Gus Wiseman, Aug 05 2023

Keywords

Examples

			The a(0) = 0 through a(5) = 18 subsets:
  .  .  .  {1,3}  {1,3}    {1,3}
           {2,3}  {1,4}    {1,4}
                  {2,3}    {1,5}
                  {3,4}    {2,3}
                  {1,3,4}  {2,5}
                  {2,3,4}  {3,4}
                           {3,5}
                           {4,5}
                           {1,2,5}
                           {1,3,4}
                           {1,3,5}
                           {1,4,5}
                           {2,3,4}
                           {2,3,5}
                           {2,4,5}
                           {3,4,5}
                           {1,3,4,5}
                           {2,3,4,5}

Crossrefs

For disjunction instead of containment we have A364463, partitions A363260.
For overlap we have A364466, partitions A364467 (strict A364536).
The complement is counted by A364671, partitions A364673, A364674, A364675.
First differences of terms are A364753, complement A364752.
Cf. A007862, A054519, A151897, A237667, A325325, A326083, A363225, A364345, A364464, A364537.

Programs

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

Formula

a(n) = 2^n - A364671(n). - Andrew Howroyd, Jan 27 2024

Extensions

a(21) onwards (using A364671) added by Andrew Howroyd, Jan 27 2024

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

A363220 Number of integer partitions of n whose conjugate has the same median.

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 3, 8, 8, 12, 12, 15, 21, 27, 36, 49, 65, 85, 112, 149, 176, 214, 257, 311, 378, 470, 572, 710, 877, 1080, 1322, 1637, 1983, 2416, 2899, 3465, 4107, 4891, 5763, 6820, 8071, 9542, 11289, 13381, 15808, 18710, 22122, 26105, 30737, 36156, 42377
Offset: 1

Views

Author

Gus Wiseman, May 29 2023

Keywords

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The partition y = (4,3,1,1) has median 2, and its conjugate (4,2,2,1) also has median 2, so y is counted under a(9).
The a(1) = 1 through a(9) = 8 partitions:
  (1)  .  (21)  (22)  (311)  (321)   (511)    (332)     (333)
                             (411)   (4111)   (422)     (711)
                             (3111)  (31111)  (611)     (4221)
                                              (3311)    (4311)
                                              (4211)    (6111)
                                              (5111)    (51111)
                                              (41111)   (411111)
                                              (311111)  (3111111)

Crossrefs

For mean instead of median we have A047993.
For product instead of median we have A325039, ranks A325040.
For union instead of conjugate we have A360245, complement A360244.
Median of conjugate by rank is A363219.
These partitions are ranked by A363261.
A000700 counts self-conjugate partitions, ranks A088902.
A046682 and A352487-A352490 pertain to excedance set.
A122111 represents partition conjugation.
A325347 counts partitions with integer median.
A330644 counts non-self-conjugate partitions (twice A000701), ranks A352486.
A352491 gives n minus Heinz number of conjugate.
Cf. A000975, A067538, A114638, A360068, A360242, A360248, A362617, A362618, A362621, A363223, A363260.

Programs

  • Mathematica
    conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],Median[#]==Median[conj[#]]&]],{n,30}]
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