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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A321929 Tetrangle where T(n,H(u),H(v)) is the coefficient of f(v) in s(u), where u and v are integer partitions of n, H is Heinz number, f is forgotten symmetric functions, and s is Schur functions.

Original entry on oeis.org

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

Links

Crossrefs

This is a regrouping of the triangle A321892.
Cf. A008480, A056239, A124794, A124795, A153452, A215366, A296188, A300121, A319191, A319193, A321912-A321935.

A321930 Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in f(u), where u and v are integer partitions of n, H is Heinz number, f is forgotten symmetric functions, and s is Schur functions.

Original entry on oeis.org

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

Links

Crossrefs

This is a regrouping of the triangle A321894.
Cf. A008480, A056239, A124794, A124795, A153452, A215366, A296188, A300121, 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.
Previous Showing 21-24 of 24 results.

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