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binomial expansion functions
EvalBinExp(B:TAGGED("$binrepr.BinExp"), Up:INT, Down:INT):INT
where N and K are positive integers, and Up and Down are integers.
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The function
BinExp
computes the K-binomial
expansion of N, i.e., the unique expression
N = Bin(N(K),K) + Bin(N(K-1),K-1) + ... + Bin(N(I),I)
where
N(K) > ... > N(I) >= 1, for some I.
This function computes the sum of the binomial coefficients
appearing in the K-binomial expansion of N after replacing each
summand Bin(N(J),J) by Bin(N(J)+Up,J+Down). It is useful in
generalizations of Macaulay's theorem characterizing Hilbert
functions.
It is the same as
BinExp
with 4 arguments except it
takes a precomputed binomial expansion as an argument rather than N and K.
BE := BinExp(13,4);
BE;
Bin(5,4) + Bin(4,3) + Bin(3,2) + Bin(1,1)
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EvalBinExp(BE,1,1);
16
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BinExp(13,4,1,1);
16
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