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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A321927 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in f(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and f is forgotten symmetric functions.

Original entry on oeis.org

1, -1, 0, 1, 1, 1, 0, 0, -2, -1, 0, 1, 1, 1, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 1, 0, 0, -3, -2, -2, -1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -2, -1, 0, 0, 0, 0, 0, -2, 0, -1, 0, 0, 0, 0, 3, 1, 2, 1, 0, 0, 0, 3, 2, 1, 0, 1, 0, 0, -4, -3, -3, -2, -2, -1, 0
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 f(v) in m(u).

Examples

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

Links

Crossrefs

This is a regrouping of the triangle A321886.
Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

A321928 Tetrangle where T(n,H(u),H(v)) is the coefficient of f(v) in p(u), where u and v are integer partitions of n, H is Heinz number, f is forgotten symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, -1, 0, 1, 2, 1, 0, 0, -1, -1, 0, 1, 3, 6, -1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 0, -1, -2, -2, -2, 0, 1, 6, 4, 12, 24, 1, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 1, 2, 1, 0, 2, 0, 0, -1, -3, -4, -6, -6, -6
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):  -1
  (11):  1  2
.
  (3):    1
  (21):  -1 -1
  (111):  1  3  6
.
  (4):    -1
  (22):    1  2
  (31):    1     1
  (211):  -1 -2 -2 -2
  (1111):  1  6  4 12 24
.
  (5):      1
  (41):    -1 -1
  (32):    -1    -1
  (221):    1  1  2  2
  (311):    1  2  1     2
  (2111):  -1 -3 -4 -6 -6 -6
  (11111):  1  5 10 30 20 60 20
For example, row 14 gives: p(32) = -f(5) - f(32).

Links

Crossrefs

An unsigned version is A321917. This is a regrouping of the triangle A321888.
Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

A321932 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in e(u) * Product_i u_i!, where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and e is elementary symmetric functions.

Original entry on oeis.org

1, -1, 1, 0, 1, 2, -3, 1, 0, -1, 1, 0, 0, 1, -6, 3, 8, -6, 1, 0, 1, 0, -2, 1, 0, 0, 2, -3, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1, 24, -30, -20, 15, 20, -10, 1, 0, -6, 0, 3, 8, -6, 1, 0, 0, -2, 3, 2, -4, 1, 0, 0, 0, 1, 0, -2, 1, 0, 0, 0, 0, 2, -3, 1, 0, 0, 0, 0, 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).

Examples

			Tetrangle begins (zeros not shown):
  (1):  1
.
  (2):  -1  1
  (11):     1
.
  (3):    2 -3  1
  (21):     -1  1
  (111):        1
.
  (4):    -6  3  8 -6  1
  (22):       1    -2  1
  (31):          2 -3  1
  (211):           -1  1
  (1111):              1
.
  (5):     24 30 20 15 20 10  1
  (41):       -6     3  8 -6  1
  (32):          -2  3  2 -4  1
  (221):             1    -2  1
  (311):                2 -3  1
  (2111):                 -1  1
  (11111):                    1
For example, row 14 gives: 12e(32) = -2p(32) + 3p(221) + 2p(311) - 4p(2111) + p(11111).

Links

Crossrefs

Row sums are A134286. This is a regrouping of the triangle A321896.
Cf. A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

A321933 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in h(u) * Product_i u_i!, where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and h is homogeneous symmetric functions.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 3, 1, 0, 1, 1, 0, 0, 1, 6, 3, 8, 6, 1, 0, 1, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 24, 30, 20, 15, 20, 10, 1, 0, 6, 0, 3, 8, 6, 1, 0, 0, 2, 3, 2, 4, 1, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 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).

Examples

			Tetrangle begins (zeros not shown):
  (1):  1
.
  (2):   1  1
  (11):     1
.
  (3):    2  3  1
  (21):      1  1
  (111):        1
.
  (4):     6  3  8  6  1
  (22):       1     2  1
  (31):          2  3  1
  (211):            1  1
  (1111):              1
.
  (5):     24 30 20 15 20 10  1
  (41):        6     3  8  6  1
  (32):           2  3  2  4  1
  (221):             1     2  1
  (311):                2  3  1
  (2111):                  1  1
  (11111):                    1
For example, row 14 gives: 12h(32) = 2p(32) + 3p(221) + 2p(311) + 4p(2111) + p(11111).

Links

Crossrefs

This is a regrouping of the triangle A321897.
Cf. A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

A322076 Number of set multipartitions (multisets of sets) with no singletons, of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 0, 0, 0, 3, 1, 0, 2, 0, 0, 1, 0, 11, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 13, 1, 1, 0, 0, 0, 7, 0, 0, 0, 0, 0, 3, 0, 0, 1, 41, 0, 0, 0, 0, 0, 1, 0, 20, 0, 0, 2, 0, 0, 0, 0, 6, 16, 0, 0, 1, 0
Offset: 1

Views

Author

Gus Wiseman, Nov 25 2018

Keywords

Comments

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Examples

			The a(90) = 7 set multipartitions of {1,1,1,2,2,3,3,4} with no singletons:
  {{1,2},{1,2},{1,3},{3,4}}
  {{1,2},{1,3},{1,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3}}
  {{1,2},{1,3},{1,2,3,4}}
  {{1,2},{1,2,3},{1,3,4}}
  {{1,3},{1,2,3},{1,2,4}}
  {{1,4},{1,2,3},{1,2,3}}

Crossrefs

Cf. A001055, A007716, A049311, A056156, A116540, A181821, A255906, A304382.
Cf. A318284, A318286, A318360, A318361, A318362, A318369.

Programs

  • Mathematica
    nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sqnopfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqnopfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],!PrimeQ[#]&&SquareFreeQ[#]&]}]];
Table[Length[sqnopfacs[Times@@Prime/@nrmptn[n]]],{n,30}]
Previous Showing 51-55 of 55 results.

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