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BinExp

binomial expansion

Syntax
BinExp(N:INT,K:INT):TAGGED("$binrepr.BinExp")
BinExp(N:INT,K:INT,Up:INT,Down:INT):INT

where N and K are positive integers, and Up and Down are integers.

Description
This function 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. The value returned is tagged for pretty printing.

It can also compute 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.

Example
BE := BinExp(13,4);
BE;
Bin(5,4) + Bin(4,3) + Bin(3,2) + Bin(1,1)
-------------------------------
BinExp(13,4,1,1);
16
------------------------------- 


See Also