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.

Showing 1-10 of 39 results. Next

A321742 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in e(u), where H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 3, 0, 0, 0, 0, 1, 1, 3, 6, 0, 1, 0, 2, 6, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 5, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 5, 0, 0, 0, 1, 0, 3, 10, 1, 6, 4, 12, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Nov 19 2018

Keywords

Comments

Row n has length A000041(A056239(n)).
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			Triangle begins:
   1
   1
   0   1
   1   2
   0   0   1
   0   1   3
   0   0   0   0   1
   1   3   6
   0   1   0   2   6
   0   0   0   1   4
   0   0   0   0   0   0   1
   0   2   1   5  12
   0   0   0   0   0   0   0   0   0   0   1
   0   0   0   0   0   1   5
   0   0   0   1   0   3  10
   1   6   4  12  24
   0   0   0   0   0   0   0   0   0   0   0   0   0   0   1
   0   0   1   5   2  12  30
For example, row 12 gives: e(211) = 2m(22) + m(31) + 5m(211) + 12m(1111).

Links

Crossrefs

Row sums are A321743.
Cf. A008480, A049311, A056239, A116540, A124794, A124795, A300121, A319193, A321738, A321742-A321765, A321854.

Programs

  • Mathematica
    primeMS[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]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Table[Sum[Times@@Factorial/@Length/@Split[Sort[Length/@mtn,Greater]]/Times@@Factorial/@Length/@Split[mtn],{mtn,Select[mps[nrmptn[n]],And[And@@UnsameQ@@@#,Sort[Length/@#]==primeMS[k]]&]}],{k,Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}],{n,18}]

A321895 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in M(u), where H is Heinz number, M is augmented monomial symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Nov 20 2018

Keywords

Comments

Row n has length A000041(A056239(n)).
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

			Triangle begins:
   1
   1
   1   0
  -1   1
   1   0   0
  -1   1   0
   1   0   0   0   0
   2  -3   1
  -1   1   0   0   0
  -1   0   1   0   0
   1   0   0   0   0   0   0
   2  -1  -2   1   0
   1   0   0   0   0   0   0   0   0   0   0
  -1   1   0   0   0   0   0
  -1   0   1   0   0   0   0
  -6   3   8  -6   1
   1   0   0   0   0   0   0   0   0   0   0   0   0   0   0
   2  -1  -2   1   0   0   0
For example, row 12 gives: M(211) = 2p(4) - p(22) - 2p(31) + p(211).

Links

Crossrefs

Row sums are A080339.
Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A319193, A321742-A321765.

Programs

  • Mathematica
    primeMS[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,_}];
Table[Sum[Product[(-1)^(Length[t]-1)*(Length[t]-1)!,{t,s}],{s,Select[sps[Range[PrimeOmega[n]]]/.Table[i->If[n==1,{},primeMS[n]][[i]],{i,PrimeOmega[n]}],Times@@Prime/@Total/@#==m&]}],{n,18},{m,Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}]

A321854 Irregular triangle where T(H(u),H(v)) is the number of ways to partition the Young diagram of u into vertical sections whose sizes are the parts of v, where H is Heinz number.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Nov 19 2018

Keywords

Comments

Row n has length A000041(A056239(n)).
A vertical section is a partial Young diagram with at most one square in each row.

Examples

			Triangle begins:
  1
  1
  0  1
  1  1
  0  0  1
  0  2  1
  0  0  0  0  1
  1  3  1
  0  2  0  4  1
  0  0  0  3  1
  0  0  0  0  0  0  1
  0  2  2  5  1
  0  0  0  0  0  0  0  0  0  0  1
  0  0  0  0  0  4  1
  0  0  0  6  0  6  1
  1  3  4  6  1
  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1
  0  0  4 10  4  8  1
The 12th row counts the following partitions of the Young diagram of (211) into vertical sections (shown as colorings by positive integers):
  T(12,7) = 0:
.
  T(12,9) = 2:    1 2   1 2
                  1     2
                  2     1
.
  T(12,10) = 2:   1 2   1 2
                  2     1
                  2     1
.
  T(12,12) = 5:   1 2   1 2   1 2   1 2   1 2
                  3     2     3     1     3
                  3     3     2     3     1
.
  T(12,16) = 1:   1 2
                  3
                  4

Crossrefs

Cf. A000085, A000110, A007016, A056239, A122111, A153452, A215366, A296188, A300121, A318396, A321719-A321731, A321737, A321738, A321742-A321765.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Rest[Subsets[ptnpos[y]]],UnsameQ@@First/@#&];
Table[With[{y=Reverse[primeMS[n]]},Table[Length[Select[spsu[ptnverts[y],ptnpos[y]],Sort[Length/@#]==primeMS[k]&]],{k,Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}]],{n,18}]

A321750 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in p(u), where H is Heinz number, m is monomial symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 3, 6, 1, 2, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 6, 4, 12, 24, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Nov 20 2018

Keywords

Comments

Row n has length A000041(A056239(n)).
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			Triangle begins:
   1
   1
   1   0
   1   2
   1   0   0
   1   1   0
   1   0   0   0   0
   1   3   6
   1   2   0   0   0
   1   0   1   0   0
   1   0   0   0   0   0   0
   1   2   2   2   0
   1   0   0   0   0   0   0   0   0   0   0
   1   1   0   0   0   0   0
   1   0   1   0   0   0   0
   1   6   4  12  24
   1   0   0   0   0   0   0   0   0   0   0   0   0   0   0
   1   1   2   2   0   0   0
For example, row 18 gives: p(221) = m(5) + 2m(32) + m(41) + 2m(221).

Links

Crossrefs

Row sums are A321751.
Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A319182, A319191, A321742-A321765.

A321752 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of e(v) in p(u), where H is Heinz number, e is elementary symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 1, -2, 1, 0, 1, 3, -3, 1, 0, -2, 1, -4, 2, 4, -4, 1, 0, 0, 1, 0, 4, 0, -4, 1, 0, 0, 3, -3, 1, 5, -5, -5, 5, 5, -5, 1, 0, 0, 0, -2, 1, -6, 6, 6, 3, -2, -6, -12, 9, 6, -6, 1, 0, -4, 0, 2, 4, -4, 1, 0, 0, -6, 6, 3, -5, 1, 0, 0, 0, 0, 1, 7, -7, -7, -7, 14, 7, 7
Offset: 1

Views

Author

Gus Wiseman, Nov 20 2018

Keywords

Comments

Row n has length A000041(A056239(n)).
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			Triangle begins:
   1
   1
  -2   1
   0   1
   3  -3   1
   0  -2   1
  -4   2   4  -4   1
   0   0   1
   0   4   0  -4   1
   0   0   3  -3   1
   5  -5  -5   5   5  -5   1
   0   0   0  -2   1
  -6   6   6   3  -2  -6 -12   9   6  -6   1
   0  -4   0   2   4  -4   1
   0   0  -6   6   3  -5   1
   0   0   0   0   1
   7  -7  -7  -7  14   7   7   7  -7  -7 -21  14   7  -7   1
   0   0   0   4   0  -4   1
For example, row 15 gives: p(32) = -6e(32) + 6e(221) + 3e(311) - 5e(2111) + e(11111).

Links

Crossrefs

Row sums are A321753.
Cf. A005651, A008480, A056239, A124794, A124795, A135278, A296150, A319193, A319225, A319226, A321742-A321765, A321854.

A321751 Sum of coefficients of monomial symmetric functions in the power sum symmetric function of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 10, 3, 2, 1, 7, 1, 2, 2, 47, 1, 6, 1, 6, 2, 2, 1, 26, 3, 2, 10, 6, 1, 6, 1, 246, 2, 2, 2, 26, 1, 2, 2, 24, 1, 5, 1, 6, 6, 2, 1, 138, 3, 6, 2, 6, 1, 23, 2, 23, 2, 2, 1, 20, 1, 2, 7, 1602, 2, 5, 1, 6, 2, 6, 1, 105, 1, 2, 6, 6, 2, 5, 1, 114
Offset: 1

Views

Author

Gus Wiseman, Nov 20 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 ordered set partitions of {1, 2, ..., A001222(n)} whose blocks, when i is replaced by the i-th prime index of n, have weakly decreasing sums.

Examples

			The sum of coefficients of p(211) = m(4) + 2m(22) + 2m(31) + 2m(211) is a(12) = 7.

Links

Crossrefs

Row sums of A321750.
Cf. A005651, A008277, A008480, A056239, A124794, A124795, A296150, A319182, A319225, A319226, A321742-A321765.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Times@@Factorial/@Length/@Split[Sort[Total/@s]],{s,sps[Range[PrimeOmega[n]]]/.Table[i->If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[i]],{i,PrimeOmega[n]}]}],{n,50}]

A321754 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of h(v) in p(u), where H is Heinz number, p is power sum symmetric functions, and h is homogeneous symmetric functions.

Original entry on oeis.org

1, 1, 2, -1, 0, 1, 3, -3, 1, 0, 2, -1, 4, -2, -4, 4, -1, 0, 0, 1, 0, 4, 0, -4, 1, 0, 0, 3, -3, 1, 5, -5, -5, 5, 5, -5, 1, 0, 0, 0, 2, -1, 6, -6, -6, -3, 2, 6, 12, -9, -6, 6, -1, 0, 4, 0, -2, -4, 4, -1, 0, 0, 6, -6, -3, 5, -1, 0, 0, 0, 0, 1, 7, -7, -7, -7, 14
Offset: 1

Views

Author

Gus Wiseman, Nov 20 2018

Keywords

Comments

Row n has length A000041(A056239(n)).
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Up to sign, same as A321752.

Examples

			Triangle begins:
   1
   1
   2  -1
   0   1
   3  -3   1
   0   2  -1
   4  -2  -4   4  -1
   0   0   1
   0   4   0  -4   1
   0   0   3  -3   1
   5  -5  -5   5   5  -5   1
   0   0   0   2  -1
   6  -6  -6  -3   2   6  12  -9  -6   6  -1
   0   4   0  -2  -4   4  -1
   0   0   6  -6  -3   5  -1
   0   0   0   0   1
   7  -7  -7  -7  14   7   7   7  -7  -7 -21  14   7  -7   1
   0   0   0   4   0  -4   1
For example, row 15 gives: p(32) = 6h(32) - 6h(221) - 3h(311) + 5h(2111) - h(11111).

Links

Crossrefs

Row sums are all equal to 1.
Cf. A005651, A008480, A056239, A124794, A124795, A135278, A319193, A319225, A319226, A321742-A321765, A321854.

A321892 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of f(v) in s(u), where H is Heinz number, f is forgotten symmetric functions, and s is Schur functions.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Nov 20 2018

Keywords

Comments

Row n has length A000041(A056239(n)).
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			Triangle begins:
   1
   1
   0   1
   1   1
   0   0   1
   0   1   2
   0   0   0   0   1
   1   1   1
   0   1   0   1   2
   0   0   0   1   3
   0   0   0   0   0   0   1
   0   1   1   2   3
   0   0   0   0   0   0   0   0   0   0   1
   0   0   0   0   0   1   4
   0   0   0   1   0   2   5
For example, row 15 gives: s(32) = f(221) + 2f(2111) + 5f(11111).

Links

Crossrefs

Row sums are A321893.
Cf. A000085, A008480, A056239, A082733, A124795, A153452, A296188, A300121, A321742-A321765.

A321897 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in h(u) * Product_i u_i!, where H is Heinz number, h is homogeneous symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 3, 1, 0, 1, 1, 6, 3, 8, 6, 1, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 2, 3, 1, 24, 30, 20, 15, 20, 10, 1, 0, 0, 0, 1, 1, 120, 90, 144, 40, 15, 90, 120, 45, 40, 15, 1, 0, 6, 0, 3, 8, 6, 1, 0, 0, 2, 3, 2, 4, 1, 0, 0, 0, 0, 1, 720, 840, 504, 420, 630
Offset: 1

Views

Author

Gus Wiseman, Nov 20 2018

Keywords

Comments

Row n has length A000041(A056239(n)).
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			Triangle begins:
    1
    1
    1    1
    0    1
    2    3    1
    0    1    1
    6    3    8    6    1
    0    0    1
    0    1    0    2    1
    0    0    2    3    1
   24   30   20   15   20   10    1
    0    0    0    1    1
  120   90  144   40   15   90  120   45   40   15    1
    0    6    0    3    8    6    1
    0    0    2    3    2    4    1
    0    0    0    0    1
  720  840  504  420  630  504  210  280  105  210  420  105   70   21    1
    0    0    0    1    0    2    1
For example, row 14 gives: 12h(41) = 6p(41) + 3p(221) + 8p(311) + 6p(2111) + p(11111).

Links

Crossrefs

Row sums are A321898.
Cf. A005651, A008480, A056239, A124794, A124795, A319193, A321742-A321765, A321896.

A321900 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in S(u), where H is Heinz number, p is power sum symmetric functions, and S is augmented Schur functions.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 2, 3, 1, -1, 0, 1, 6, 3, 8, 6, 1, 2, -3, 1, 0, 3, -4, 0, 1, -2, -1, 0, 2, 1, 24, 30, 20, 15, 20, 10, 1, 2, -1, 0, -2, 1, 120, 90, 144, 40, 15, 90, 120, 45, 40, 15, 1, -6, 0, -5, 0, 5, 5, 1, 0, -6, 4, 3, -4, 2, 1, -6, 3, 8, -6, 1, 720, 840
Offset: 1

Views

Author

Gus Wiseman, Nov 20 2018

Keywords

Comments

Row n has length A000041(A056239(n)).
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
We define the augmented Schur functions to be S(y) = |y|! * s(y) / syt(y), where s is Schur functions and syt(y) is the number of standard Young tableaux of shape y.

Examples

			Triangle begins:
    1
    1
    1    1
   -1    1
    2    3    1
   -1    0    1
    6    3    8    6    1
    2   -3    1
    0    3   -4    0    1
   -2   -1    0    2    1
   24   30   20   15   20   10    1
    2   -1    0   -2    1
  120   90  144   40   15   90  120   45   40   15    1
   -6    0   -5    0    5    5    1
    0   -6    4    3   -4    2    1
   -6    3    8   -6    1
  720  840  504  420  630  504  210  280  105  210  420  105   70   21    1
    0    6   -4    3   -4   -2    1
For example, row 15 gives: S(32) = 4p(32) - 6p(41) + 3p(221) - 4p(311) + 2p(2111) + p(11111).

Links

Crossrefs

Row sums are above.
Cf. A000085, A008480, A056239, A082733, A124795, A153452, A296188, A296561, A300121, A304438, A317552, A317554, A321742-A321765.
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