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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A321931 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in M(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and M is augmented monomial symmetric functions.

Original entry on oeis.org

1, 1, 0, -1, 1, 1, 0, 0, -1, 1, 0, 2, -3, 1, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 1, 0, 0, 2, -1, -2, 1, 0, -6, 3, 8, -6, 1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 2, -1, -2, 1, 0, 0, 0, 2, -2, -1, 0, 1, 0, 0, -6, 6, 5, -3, -3, 1, 0
Offset: 1

Views

Author

Gus Wiseman, Nov 23 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
The augmented monomial symmetric functions are given by M(y) = c(y) * m(y) where c(y) = Product_i (y)_i! where (y)_i is the number of i's in y and m is monomial symmetric functions.

Examples

			Tetrangle begins (zeros not shown):
  (1):  1
.
  (2):   1
  (11): -1  1
.
  (3):    1
  (21):  -1  1
  (111):  2 -3  1
.
  (4):     1
  (22):   -1  1
  (31):   -1     1
  (211):   2 -1 -2  1
  (1111): -6  3  8 -6  1
.
  (5):      1
  (41):    -1  1
  (32):    -1     1
  (221):    2 -1 -2  1
  (311):    2 -2 -1     1
  (2111):  -6  6  5 -3 -3  1
  (11111): 24 30 20 15 20 10  1
For example, row 14 gives: M(32) = -p(5) + p(32).

Links

Crossrefs

Row sums are A155972. This is a regrouping of the triangle A321895.
Cf. A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

A337072 Number of factorizations of the superprimorial A006939(n) into squarefree numbers > 1.

Original entry on oeis.org

1, 1, 2, 10, 141, 6769, 1298995, 1148840085, 5307091649182, 143026276277298216, 24801104674619158730662, 30190572492693121799801655311, 278937095127086600900558327826721594
Offset: 0

Views

Author

Gus Wiseman, Aug 15 2020

Keywords

Comments

The n-th superprimorial is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1), which has n! divisors.
Also the number of set multipartitions (multisets of sets) of the multiset of prime factors of the superprimorial A006939(n).

Examples

			The a(1) = 1 through a(3) = 10 factorizations:
    2  2*6    2*6*30
       2*2*3  6*6*10
              2*5*6*6
              2*2*3*30
              2*2*6*15
              2*3*6*10
              2*2*3*5*6
              2*2*2*3*15
              2*2*3*3*10
              2*2*2*3*3*5
The a(1) = 1 through a(3) = 10 set multipartitions:
     {1}  {1}{12}    {1}{12}{123}
          {1}{1}{2}  {12}{12}{13}
                     {1}{1}{12}{23}
                     {1}{1}{2}{123}
                     {1}{2}{12}{13}
                     {1}{3}{12}{12}
                     {1}{1}{1}{2}{23}
                     {1}{1}{2}{2}{13}
                     {1}{1}{2}{3}{12}
                     {1}{1}{1}{2}{2}{3}

Crossrefs

A000142 counts divisors of superprimorials.
A022915 counts permutations of the same multiset.
A103774 is the version for factorials instead of superprimorials.
A337073 is the strict case (strict factorizations into squarefree numbers).
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A045778 counts strict factorizations.
A050320 counts factorizations into squarefree numbers.
A050326 counts strict factorizations into squarefree numbers.
A076954 can be used instead of A006939 (cf. A307895, A325337).
A089259 counts set multipartitions of integer partitions.
A116540 counts normal set multipartitions.
A317829 counts factorizations of superprimorials.
A337069 counts strict factorizations of superprimorials.
Cf. A000178, A002110, A027423, A124010, A181818, A303279, A318360, A336417, A337070.

Programs

  • Mathematica
    chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
facsqf[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsqf[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[facsqf[chern[n]]],{n,0,3}]
  • PARI
    \\ See A318360 for count.
a(n) = {if(n==0, 1, count(vector(n,i,i)))} \\ Andrew Howroyd, Aug 31 2020

Formula

a(n) = A050320(A006939(n)).
a(n) = A318360(A002110(n)). - Andrew Howroyd, Aug 31 2020

Extensions

a(7)-a(12) from Andrew Howroyd, Aug 31 2020

A318370 Number of non-isomorphic strict set multipartitions (sets of sets) of the multiset of prime indices of n.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 3, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 3, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 3, 1, 2, 1, 0, 2, 3, 1, 1, 2, 3, 1, 0, 1, 2, 1, 1, 2, 3, 1, 0, 0, 2, 1, 3, 2, 2, 2
Offset: 1

Views

Author

Gus Wiseman, Aug 24 2018

Keywords

Examples

			Non-isomorphic representatives of the a(180) = 4 strict set multipartitions of {1,1,2,2,3}:
  {{1,2},{1,2,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{2,3}}
  {{1},{2},{3},{1,2}}

Crossrefs

Cf. A007716, A045778, A049311, A050326, A056239, A112798, A116539, A116540, A292432, A292444, A293511, A318284, A318360, A318369.

A371455 Numbers k such that if we take the binary indices of each prime index of k we get an antichain of sets.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 35, 36, 37, 38, 41, 42, 43, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 61, 63, 64, 65, 67, 69, 71, 72, 73, 74, 76, 79, 81, 83, 84, 86, 89, 95, 96, 97, 98, 99
Offset: 1

Views

Author

Gus Wiseman, Apr 01 2024

Keywords

Comments

In an antichain of sets, no edge is a proper subset of any other.

Examples

			The prime indices of 65 are {3,6} with binary indices {{1,2},{2,3}} so 65 is in the sequence.
The prime indices of 255 are {2,3,7} with binary indices {{2},{1,2},{1,2,3}} so 255 is not in the sequence.

Links

Crossrefs

Contains all powers of primes A000961.
An opposite version is A087086, carry-connected case A371294.
For prime indices of prime indices we have A316476, carry-connected A329559.
These antichains are counted by A325109.
For binary indices of binary indices we have A326704, carry-conn. A326750.
The carry-connected case is A371445, counted by A371446.
A048143 counts connected antichains of sets.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A050320 counts set multipartitions of prime indices, see also A318360.
A070939 gives length of binary expansion.
A089259 counts set multipartitions of integer partitions.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A116540 counts normal set multipartitions.
A302478 ranks set multipartitions, cf. A073576.
A325118 ranks carry-connected partitions, counted by A325098.
A371451 counts carry-connected components of binary indices.
Cf. A019565, A050326, A304713, A305148, A325097, A325107, A371291, A371452.

Programs

  • Mathematica
    stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],stableQ[bix/@prix[#],SubsetQ]&]

A321184 Number of integer partitions of n that are the vertex-degrees of some multiset of nonempty sets, none of which is a proper subset of any other, with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 7, 6, 15, 15, 30
Offset: 0

Views

Author

Gus Wiseman, Oct 29 2018

Keywords

Examples

			The a(2) = 1 through a(8) = 15 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)
               (211)   (11111)  (222)     (3211)     (332)
               (1111)           (321)     (22111)    (422)
                                (2211)    (31111)    (431)
                                (3111)    (211111)   (2222)
                                (21111)   (1111111)  (3221)
                                (111111)             (3311)
                                                     (4211)
                                                     (22211)
                                                     (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
The a(6) = 7 integer partitions together with a realizing multi-antichain of each (the parts of the partition count the appearances of each vertex in the multi-antichain):
      (33): {{1,2},{1,2},{1,2}}
     (321): {{1,2},{1,2},{1,3}}
    (3111): {{1,2},{1,3},{1,4}}
     (222): {{1,2,3},{1,2,3}}
    (2211): {{1,2,3},{1,2,4}}
   (21111): {{1,2},{1,3,4,5}}
  (111111): {{1,2,3,4,5,6}}

Crossrefs

Cf. A000070, A000569, A006126, A096827, A147878, A209816, A283877, A318360, A319719, A319721, A320799, A320921, A321176.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
multanti[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,Min@@Length/@#>1,stableQ[#]]&];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[n],multanti[#]!={}&]],{n,8}]

A321743 Sum of coefficients of monomial symmetric functions in the elementary symmetric function of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 1, 10, 9, 5, 1, 20, 1, 6, 14, 47, 1, 50, 1, 30, 20, 7, 1, 110, 29, 8, 157, 42, 1, 97, 1, 246, 27, 9, 49, 338, 1, 10, 35, 206, 1, 159, 1, 56, 353, 11, 1, 732, 99, 224, 44, 72, 1, 1184, 76, 332, 54, 12, 1, 743, 1, 13, 677, 1602, 111, 242, 1, 90
Offset: 1

Views

Author

Gus Wiseman, Nov 19 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also the number of size-preserving permutations of set multipartitions (multisets of sets) of a multiset (such as row n of A305936) whose multiplicities are the prime indices of n.

Examples

			The sum of coefficients of e(211) = 2m(22) + m(31) + 5m(211) + 12m(1111) is a(12) = 20.
The a(2) = 1 through a(9) = 9 size-preserving permutations of set multipartitions:
  {1} {1}{1} {12}   {1}{1}{1} {1}{12}   {1}{1}{1}{1} {123}     {12}{12}
             {1}{2}           {1}{1}{2}              {1}{23}   {1}{2}{12}
             {2}{1}           {1}{2}{1}              {2}{13}   {2}{1}{12}
                              {2}{1}{1}              {3}{12}   {1}{1}{2}{2}
                                                     {1}{2}{3} {1}{2}{1}{2}
                                                     {1}{3}{2} {1}{2}{2}{1}
                                                     {2}{1}{3} {2}{1}{1}{2}
                                                     {2}{3}{1} {2}{1}{2}{1}
                                                     {3}{1}{2} {2}{2}{1}{1}
                                                     {3}{2}{1}

Crossrefs

Row sums of A321742.
Cf. A005651, A008480, A049311, A056239, A116540, A124794, A124795, A181821, A296150, A318360, A319193, A319225, A319226, A321738, A321742-A321765.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Sum[Times@@Factorial/@Length/@Split[Sort[Length/@mtn,Greater]]/Times@@Factorial/@Length/@Split[mtn],{mtn,Select[mps[nrmptn[n]],And@@UnsameQ@@@#&]}],{n,30}]

A321934 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in F(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and F is augmented forgotten symmetric functions.

Original entry on oeis.org

1, -1, 0, 1, 1, 1, 0, 0, -1, -1, 0, 2, 3, 1, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, -2, -1, -2, -1, 0, 6, 3, 8, 6, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 2, 1, 2, 1, 0, 0, 0, 2, 2, 1, 0, 1, 0, 0, -6, -6, -5, -3, -3, -1, 0
Offset: 1

Views

Author

Gus Wiseman, Nov 23 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
The augmented forgotten symmetric functions are given by F(y) = c(y) * f(y) where f is forgotten symmetric functions and c(y) = Product_i (y)_i!, where (y)_i is the number of i's in y.

Examples

			Tetrangle begins (zeros not shown):
  (1):  1
.
  (2):  -1
  (11):  1  1
.
  (3):    1
  (21):  -1 -1
  (111):  2  3  1
.
  (4):    -1
  (22):    1  1
  (31):    1     1
  (211):  -2 -1 -2 -1
  (1111):  6  3  8  6  1
.
  (5):      1
  (41):    -1 -1
  (32):    -1    -1
  (221):    2  1  2  1
  (311):    2  2  1     1
  (2111):  -6 -6 -5 -3 -3 -1
  (11111): 24 30 20 15 20 10  1
For example, row 14 gives: F(32) = -p(5) - p(32).

Links

Crossrefs

Row sums are A178803. Up to sign, same as A321931. This is a regrouping of the triangle A321899.
Cf. A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

A383308 Number of integer partitions of n that can be partitioned into sets with a common sum.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 6, 10, 13, 15, 13, 31
Offset: 0

Views

Author

Gus Wiseman, Apr 25 2025

Keywords

Comments

Any strict partition can be partitioned into a single set, so we have a lower bound a(n) >= A000009(n).

Examples

			The multiset (3,2,2,1,1) has partition {{3},{1,2},{1,2}}, so is counted under a(9).
The a(1) = 1 through a(9) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)         (9)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)        (54)
             (111)  (31)    (41)     (42)      (52)       (53)        (63)
                    (1111)  (11111)  (51)      (61)       (62)        (72)
                                     (222)     (421)      (71)        (81)
                                     (321)     (1111111)  (431)       (333)
                                     (2211)               (521)       (432)
                                     (111111)             (2222)      (531)
                                                          (3311)      (621)
                                                          (11111111)  (3321)
                                                                      (32211)
                                                                      (222111)
                                                                      (111111111)

Crossrefs

Twice-partitions of this type (into sets with a common sum) are counted by A279788.
Multiset partitions of this type are ranked by A326534 /\ A302478.
For distinct instead of equal sums we have A381992, see also A382077.
The complement is counted by A381994, ranks A381719.
Partitions of prime indices of this type are counted by A382080.
Normal multiset partitions of this type are counted by A382429, see A326518.
For constant instead of strict blocks we have A383093, ranks A383014.
A000041 counts integer partitions, strict A000009.
A001055 counts factorizations, strict A045778.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
Cf. A050320, A293511, A321455, A358914, A381633, A381717, A381993.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[IntegerPartitions[n],Length[Select[mps[#],And@@UnsameQ@@@#&&SameQ@@Total/@#&]]>0&]],{n,0,10}]

A321916 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in h(u), where u and v are integer partitions of n, H is Heinz number, e is elementary symmetric functions, and h is homogeneous symmetric functions.

Original entry on oeis.org

1, -1, 1, 0, 1, 1, -2, 1, 0, -1, 1, 0, 0, 1, -1, 1, 2, -3, 1, 0, 1, 0, -2, 1, 0, 0, 1, -2, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1, 1, -2, -2, 3, 3, -4, 1, 0, -1, 0, 1, 2, -3, 1, 0, 0, -1, 2, 1, -3, 1, 0, 0, 0, 1, 0, -2, 1, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, -1, 1
Offset: 1

Views

Author

Gus Wiseman, Nov 22 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also the coefficient of h(v) in e(u).

Examples

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):  -1  1
  (11):     1
.
  (3):    1 -2  1
  (21):     -1  1
  (111):        1
.
  (4):    -1  1  2 -3  1
  (22):       1    -2  1
  (31):          1 -2  1
  (211):           -1  1
  (1111):              1
.
  (5):      1 -2 -2  3  3 -4  1
  (41):       -1     1  2 -3  1
  (32):          -1  2  1 -3  1
  (221):             1    -2  1
  (311):                1 -2  1
  (2111):                 -1  1
  (11111):                    1
For example, row 14 gives: h(32) = -e(32) + 2e(221) + e(311) - 3e(2111) + e(11111).

Links

Crossrefs

This is a regrouping of the triangle A321749. Row sums are A134286.
Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

A321919 Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in p(u), where u and v are integer partitions of n, H is Heinz number, h is homogeneous symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 2, -1, 0, 1, 3, -3, 1, 0, 2, -1, 0, 0, 1, 4, -2, -4, 4, -1, 0, 4, 0, -4, 1, 0, 0, 3, -3, 1, 0, 0, 0, 2, -1, 0, 0, 0, 0, 1, 5, -5, -5, 5, 5, -5, 1, 0, 4, 0, -2, -4, 4, -1, 0, 0, 6, -6, -3, 5, -1, 0, 0, 0, 4, 0, -4, 1, 0, 0, 0, 0, 3, -3, 1, 0, 0, 0, 0, 0, 2
Offset: 1

Views

Author

Gus Wiseman, Nov 22 2018

Keywords

Comments

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

Examples

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):   2 -1
  (11):     1
.
  (3):    3 -3  1
  (21):      2 -1
  (111):        1
.
  (4):     4 -2 -4  4 -1
  (22):       4    -4  1
  (31):          3 -3  1
  (211):            2 -1
  (1111):              1
.
  (5):      5 -5 -5  5  5 -5  1
  (41):        4    -2 -4  4 -1
  (32):           6 -6 -3  5 -1
  (221):             4    -4  1
  (311):                3 -3  1
  (2111):                  2 -1
  (11111):                    1
For example, row 14 gives: p(32) = 6h(32) - 6h(221) - 3h(311) + 5h(2111) - h(11111).

Links

Crossrefs

This is a regrouping of the triangle A321754.
Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.
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