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allocated for Gus WisemanIrregular 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.

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

1,3

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.

Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_polynomial">Symmetric polynomial</a>

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).

Row sums are A080339.

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

allocated

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Gus Wiseman, Nov 20 2018

approved

editing

allocated for Gus Wiseman

allocated

approved