# CoCoA:BinExp

From ApCoCoAWiki

## BinExp

binomial expansion

### Description

This function computes the K-binomial expansion of N, i.e., the

unique expression

<verbatim>

N = Bin(N(K),K) + Bin(N(K-1),K-1) + ... + Bin(N(I),I)

</verbatim> where <formula>N(K) > ... > N(I) >= 1</formula>, 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 -------------------------------

### Syntax

BinExp(N:INT,K:INT):TAGGED(<quotes>$binrepr.BinExp</quotes>) BinExp(N:INT,K:INT,Up:INT,Down:INT):INT where N and K are positive integers, and Up and Down are integers.

<type>integer</type>