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A321900
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Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in S(u), where H is Heinz number, p is power sum symmetric functions, and S is augmented Schur functions.
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3
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1, 1, 1, 1, -1, 1, 2, 3, 1, -1, 0, 1, 6, 3, 8, 6, 1, 2, -3, 1, 0, 3, -4, 0, 1, -2, -1, 0, 2, 1, 24, 30, 20, 15, 20, 10, 1, 2, -1, 0, -2, 1, 120, 90, 144, 40, 15, 90, 120, 45, 40, 15, 1, -6, 0, -5, 0, 5, 5, 1, 0, -6, 4, 3, -4, 2, 1, -6, 3, 8, -6, 1, 720, 840
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OFFSET
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1,7
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COMMENTS
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The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
We define the augmented Schur functions to be S(y) = |y|! * s(y) / syt(y), where s is Schur functions and syt(y) is the number of standard Young tableaux of shape y.
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LINKS
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EXAMPLE
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Triangle begins:
1
1
1 1
-1 1
2 3 1
-1 0 1
6 3 8 6 1
2 -3 1
0 3 -4 0 1
-2 -1 0 2 1
24 30 20 15 20 10 1
2 -1 0 -2 1
120 90 144 40 15 90 120 45 40 15 1
-6 0 -5 0 5 5 1
0 -6 4 3 -4 2 1
-6 3 8 -6 1
720 840 504 420 630 504 210 280 105 210 420 105 70 21 1
0 6 -4 3 -4 -2 1
For example, row 15 gives: S(32) = 4p(32) - 6p(41) + 3p(221) - 4p(311) + 2p(2111) + p(11111).
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CROSSREFS
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Row sums are above.
Cf. A000085, A008480, A056239, A082733, A124795, A153452, A296188, A296561, A300121, A304438, A317552, A317554, A321742-A321765.
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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