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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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}]

A321747 Sum of coefficients of elementary symmetric functions in the monomial symmetric function of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, -1, 1, 1, -2, -1, 1, 1, 2, 1, -3, -1, -2, -2, 1, 1, 3, -1, 3, 2, 2, 1, -4, 1, -2, -1, -3, -1, -6, 1, 1, -2, 2, -2, 6, -1, -2, 2, 4, 1, 6, -1, 3, 3, 2, 1, -5, 1, 3, -2, -3, -1, -4, 2, -4, 2, -2, 1, -12, -1, 2, -3, 1, -2, -6, 1, 3, -2, -6, -1, 10, 1, -2
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).

Examples

			The sum of coefficients of m(2211) = 9e(6) + e(42) - 4e(51) is a(36) = 6.

Links

Crossrefs

Row sums of A321746. An unsigned version is A008480.
Cf. A005651, A056239, A124794, A124795, A296150, A321738, A321742-A321765.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[(-1)^(Total[primeMS[n]]-PrimeOmega[n])*Length[Permutations[primeMS[n]]],{n,50}]

Formula

a(n) = (-1)^(A056239(n) - A001222(n)) * A008480(n).

A321748 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of h(v) in m(u), where H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.

Original entry on oeis.org

1, 1, 2, -1, -1, 1, 3, -3, 1, -3, 5, -2, 4, -2, -4, 4, -1, 1, -2, 1, -2, 3, 2, -4, 1, -4, 2, 7, -7, 2, 5, -5, -5, 5, 5, -5, 1, 4, -4, -7, 10, -3, 6, -6, -6, -3, 2, 6, 12, -9, -6, 6, -1, -5, 9, 5, -7, -9, 9, -2, -5, 5, 11, -11, -8, 10, -2, -1, 1, 2, -3, 1, 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).
Also the coefficient of e(v) in f(u), where e is elementary symmetric functions and f is forgotten symmetric functions.

Examples

			Triangle begins:
   1
   1
   2  -1
  -1   1
   3  -3   1
  -3   5  -2
   4  -2  -4   4  -1
   1  -2   1
  -2   3   2  -4   1
  -4   2   7  -7   2
   5  -5  -5   5   5  -5   1
   4  -4  -7  10  -3
   6  -6  -6  -3   2   6  12  -9  -6   6  -1
  -5   9   5  -7  -9   9  -2
  -5   5  11 -11  -8  10  -2
  -1   1   2  -3   1
   7  -7  -7  -7  14   7   7   7  -7  -7 -21  14   7  -7   1
   5  -7 -11  14  10 -14   3
For example, row 10 gives: m(31) = -4h(4) + 2h(22) + 7h(31) - 7h(211) + 2h(1111).

Links

Crossrefs

Row sums are A080339.
Cf. A005651, A008480, A048994, A056239, A124794, A124795, A135278, A300121, A319191, A319193, A321738, A321742-A321765.
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