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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A321915 Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in m(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.

Original entry on oeis.org

1, 2, -1, -1, 1, 3, -3, 1, -3, 5, -2, 1, -2, 1, 4, -2, -4, 4, -1, -2, 3, 2, -4, 1, -4, 2, 7, -7, 2, 4, -4, -7, 10, -3, -1, 1, 2, -3, 1, 5, -5, -5, 5, 5, -5, 1, -5, 9, 5, -7, -9, 9, -2, -5, 5, 11, -11, -8, 10, -2, 5, -7, -11, 14, 10, -14, 3, 5, -9, -8, 10, 12
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 e(v) in f(u), where f is forgotten symmetric functions and e is elementary symmetric functions.

Examples

			Tetrangle begins:
  (1):  1
.
  (2):   2 -1
  (11): -1  1
.
  (3):    3 -3  1
  (21):  -3  5 -2
  (111):  1 -2  1
.
  (4):     4 -2 -4  4 -1
  (22):   -2  3  2 -4  1
  (31):   -4  2  7 -7  2
  (211):   4 -4 -7 10 -3
  (1111): -1  1  2 -3  1
.
  (5):      5 -5 -5  5  5 -5  1
  (41):    -5  9  5 -7 -9  9 -2
  (32):    -5  5 11 11 -8 10 -2
  (221):    5 -7 11 14 10 14  3
  (311):    5 -9 -8 10 12 13  3
  (2111):  -5  9 10 14 13 17 -4
  (11111):  1 -2 -2  3  3 -4  1
For example, row 14 gives: m(32) = -5h(5) + 11h(32) + 5h(41) - 11h(221) - 8h(311) + 10h(2111) - 2h(11111).

Links

Crossrefs

This is a regrouping of the triangle A321748. Row sums are A155972.
Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A319191, A319193, A321912-A321935.

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.

A321920 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in s(u), where u and v are integer partitions of n, H is Heinz number, e is elementary symmetric functions, and s is Schur functions.

Original entry on oeis.org

1, -1, 1, 1, 0, 1, -2, 1, -1, 1, 0, 1, 0, 0, -1, 1, 2, -3, 1, 0, 1, -1, 0, 0, 1, -1, -1, 1, 0, -1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, -2, -2, 3, 3, -4, 1, -1, 1, 2, -2, -1, 1, 0, 0, 1, -1, 1, -1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, -1, 0, 1, 0, 0, -1, 1, 0, 0, 0, 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).

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 -1
  (31):    1 -1 -1  1
  (211):  -1     1
  (1111):  1
.
  (5):      1 -2 -2  3  3 -4  1
  (41):    -1  1  2 -2 -1  1
  (32):        1 -1  1 -1
  (221):      -1  1
  (311):    1 -1 -1     1
  (2111):  -1  1
  (11111):  1
For example, row 14 gives: s(32) = -e(32) + e(41) + e(221) - e(311).

Links

Crossrefs

Row sums are A134286. This is a regrouping of the triangle A321755.
Cf. A005651, A008480, A056239, A124794, A124795, A153452, A215366, A296188, A300121, A319191, A319193, A321912-A321935.

A321921 Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in e(u), where u and v are integer partitions of n, H is Heinz number, s is Schur functions, and e is elementary symmetric functions.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 2, 1, 1, 2, 3, 3, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 2, 1, 2, 1, 0, 0, 0, 1, 1, 2, 1, 0, 1, 2, 3, 3, 3, 1, 1, 4, 5, 5, 6, 4
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 1
.
  (3):       1
  (21):    1 1
  (111): 1 2 1
.
  (4):            1
  (22):     1   1 1
  (31):         1 1
  (211):    1 1 2 1
  (1111): 1 2 3 3 1
.
  (5):                 1
  (41):              1 1
  (32):          1   1 1
  (221):       1 2 1 2 1
  (311):         1 1 2 1
  (2111):    1 2 3 3 3 1
  (11111): 1 4 5 5 6 4 1
For example, row 14 gives: e(32) = s(221) + s(2111) + s(11111).

Links

Crossrefs

This is a regrouping of the triangle A321756.
Cf. A005651, A008480, A056239, A124794, A124795, A153452, A215366, A296188, A300121, A319191, A319193, A321912-A321935.

A321922 Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in s(u), where u and v are integer partitions of n, H is Heinz number, h is homogeneous symmetric functions, and s is Schur functions.

Original entry on oeis.org

1, 1, 0, -1, 1, 1, 0, 0, -1, 1, 0, 1, -2, 1, 1, 0, 0, 0, 0, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, -1, 1, 0, -1, 1, 2, -3, 1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 1, -1, 1, -1, 0, 0, 1, -1, -1, 0, 1, 0, 0, -1, 1, 2, -2, -1, 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).

Examples

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

Links

Crossrefs

Row sums are A155972. This is a regrouping of the triangle A321758.
Cf. A005651, A008480, A056239, A124794, A124795, A153452, A215366, A296188, A300121, A319191, A319193, A321912-A321935.

A321923 Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in h(u), where u and v are integer partitions of n, H is Heinz number, s is Schur functions, and h is homogeneous symmetric functions.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 2, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 2, 1, 0, 1, 2, 3, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 2, 2, 1, 1, 0, 0, 1, 2, 1, 0, 1, 0, 0, 1, 3, 3, 2, 3, 1, 0, 1, 4, 5, 5, 6, 4
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 1
.
  (3):   1
  (21):  1 1
  (111): 1 2 1
.
  (4):    1
  (22):   1 1 1
  (31):   1   1
  (211):  1 1 2 1
  (1111): 1 2 3 3 1
.
  (5):     1
  (41):    1 1
  (32):    1 1 1
  (221):   1 2 2 1 1
  (311):   1 2 1   1
  (2111):  1 3 3 2 3 1
  (11111): 1 4 5 5 6 4 1
For example, row 14 gives: h(32) = s(5) + s(32) + s(41).

Links

Crossrefs

This is a regrouping of the triangle A321759.
Cf. A005651, A008480, A056239, A124794, A124795, A153452, A215366, A296188, A300121, A319191, A319193, A321912-A321935.

A321926 Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in p(u), where u and v are integer partitions of n, H is Heinz number, s is Schur functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 1, -1, 1, 1, 1, -1, 1, 1, 0, -1, 1, 2, 1, 1, 0, -1, 1, -1, 1, 2, -1, -1, 1, 1, -1, 0, 0, 1, 1, 0, 1, -1, -1, 1, 2, 3, 3, 1, 1, -1, 0, 0, 1, -1, 1, 1, 0, -1, 1, 0, 0, -1, 1, -1, 1, -1, 0, 1, -1, 1, 0, 1, 1, -2, 0, 1, 1, 1, -1, -1, 0, 1, 1, 1, 2, 1, -1, 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):   1 -1
  (11):  1  1
.
  (3):    1 -1  1
  (21):   1    -1
  (111):  1  2  1
.
  (4):     1    -1  1 -1
  (22):    1  2 -1 -1  1
  (31):    1 -1        1
  (211):   1     1 -1 -1
  (1111):  1  2  3  3  1
.
  (5):      1 -1        1 -1  1
  (41):     1    -1  1       -1
  (32):     1 -1  1 -1     1 -1
  (221):    1     1  1 -2     1
  (311):    1  1 -1 -1     1  1
  (2111):   1  2  1 -1    -2 -1
  (11111):  1  4  5  5  6  4  1
For example, row 14 gives: p(32) = s(5) + s(32) - s(41) - s(221) + s(2111) - s(11111).

Links

Crossrefs

Row sums are A317552. This is a regrouping of the triangle A321765.
Cf. A005651, A008480, A056239, A124794, A124795, A153452, A215366, A296188, A300121, A319191, A319193, A321912-A321935.

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.
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