CoCoA:EvalBinExp

EvalBinExp

binomial expansion functions

Description

The function <ttref>BinExp</ttref> 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.

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 <ttref>BinExp</ttref> with 4 arguments except it

takes a precomputed binomial expansion as an argument rather than N and K.

Example

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

Syntax

EvalBinExp(B:TAGGED(<quotes>$binrepr.BinExp</quotes>),Up:INT,Down:INT):INT

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

Bin

BinExp

   <type>integer</type>