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 23 results. Next

A321912 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in e(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

Original entry on oeis.org

1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 3, 1, 3, 6, 0, 0, 0, 0, 1, 0, 1, 0, 2, 6, 0, 0, 0, 1, 4, 0, 2, 1, 5, 12, 1, 6, 4, 12, 24, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 5, 0, 0, 0, 1, 0, 3, 10, 0, 0, 1, 5, 2, 12, 30, 0, 0, 0, 2, 1, 7, 20, 0, 1, 3, 12, 7, 27, 60, 1, 5
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 h(u), where f is forgotten symmetric functions and h is homogeneous symmetric functions.

Examples

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):      1
  (11):  1  2
.
  (3):          1
  (21):      1  3
  (111):  1  3  6
.
  (4):                 1
  (22):       1     2  6
  (31):             1  4
  (211):      2  1  5 12
  (1111):  1  6  4 12 24
.
  (5):                        1
  (41):                    1  5
  (32):              1     3 10
  (221):          1  5  2 12 30
  (311):             2  1  7 20
  (2111):      1  3 12  7 27 60
  (11111):  1  5 10 30 20 60 20
For example, row 14 gives: e(32) = m(221) + 3m(2111) + 10m(11111).

Links

Crossrefs

This is a regrouping of the triangle A321742.
Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321854, A321738, A321913-A321935.

A321914 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in m(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

Original entry on oeis.org

1, -2, 1, 1, 0, 3, -3, 1, -3, 1, 0, 1, 0, 0, -4, 2, 4, -4, 1, 2, 1, -2, 0, 0, 4, -2, -1, 1, 0, -4, 0, 1, 0, 0, 1, 0, 0, 0, 0, 5, -5, -5, 5, 5, -5, 1, -5, 1, 5, -3, -1, 1, 0, -5, 5, -1, 1, -2, 0, 0, 5, -3, 1, 0, 0, 0, 0, 5, -1, -2, 0, 1, 0, 0, -5, 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).
Also the coefficient of h(v) in f(u), where f is forgotten symmetric functions and h is homogeneous symmetric functions.

Examples

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

Links

Crossrefs

This is a regrouping of the triangle A321746.
Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321854, A321738, A321912-A321935.

A321918 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in p(u), where u and v are integer partitions of n, H is Heinz number, e is elementary 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, 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).

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) = -6e(32) + 6e(221) + 3e(311) - 5e(2111) + e(11111).

Links

Crossrefs

This is a regrouping of the triangle A321752.
Cf. A005651, A008480, A056239, A124794, A124795, A135278, A215366, A318284, A319191, A319193, A319225, A319226, A321912-A321935.

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.

A321913 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in h(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, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 3, 6, 1, 1, 1, 1, 1, 1, 3, 2, 4, 6, 1, 2, 2, 3, 4, 1, 4, 3, 7, 12, 1, 6, 4, 12, 24, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 1, 2, 3, 5, 4, 7, 10, 1, 3, 5, 11, 8, 18, 30, 1, 3, 4, 8, 7, 13, 20, 1, 4, 7, 18, 13, 33, 60, 1, 5
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 e(u), where f is forgotten symmetric functions and e is elementary symmetric functions.

Examples

			Tetrangle begins:
  (1):  1
.
  (2):   1  1
  (11):  1  2
.
  (3):    1  1  1
  (21):   1  2  3
  (111):  1  3  6
.
  (4):     1  1  1  1  1
  (22):    1  3  2  4  6
  (31):    1  2  2  3  4
  (211):   1  4  3  7 12
  (1111):  1  6  4 12 24
.
  (5):      1  1  1  1  1  1  1
  (41):     1  2  2  3  3  4  5
  (32):     1  2  3  5  4  7 10
  (221):    1  3  5 11  8 18 30
  (311):    1  3  4  8  7 13 20
  (2111):   1  4  7 18 13 33 60
  (11111):  1  5 10 30 20 60 20
For example, row 14 gives: h(32) = m(5) + 3m(32) + 2m(41) + 5m(221) + 4m(311) + 7m(2111) + 10m(11111).

Links

Crossrefs

This is a regrouping of the triangle A321744.
Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A319191, A319193, A321912-A321935.

A321917 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in p(u), where u and v are integer partitions of n, H is Heinz number, m is monomial 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, 0, 1, 5, 10, 30
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) = m(5) + m(32).

Links

Crossrefs

This is a regrouping of the triangle A321750.
Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A319191, A319193, A321912-A321935.

A321924 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in s(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and s is Schur functions.

Original entry on oeis.org

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

Links

Crossrefs

This is a regrouping of the triangle A321761.
Cf. A005651, A008480, A056239, A124794, A124795, A153452, A215366, A296188, A300121, A319191, A319193, A321912-A321935.

A321925 Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in m(u), where u and v are integer partitions of n, H is Heinz number, s is Schur functions, and m is monomial symmetric functions.

Original entry on oeis.org

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

Examples

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

Links

Crossrefs

This is a regrouping of the triangle A321763.
Cf. A005651, A008480, A056239, A124794, A124795, A153452, A215366, A296188, A300121, A319191, A319193, A321912-A321935.

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.

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.
Showing 1-10 of 23 results. Next

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