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BinomialRepr, BinomialReprShift

binomial representation of integers

Syntax 
BinomialRepr(N: INT, K: INT): LIST of INT
BinomialReprShift(N: INT, K: INT, Up: INT, Down: INT): INT

where N and K are positive.

Description
The function BinomialRepr computes the K -binomial representation of N , also called Macaulay representation, i.e. the unique expression 
  N = binomial(N(K),K) + binomial(N(K-1),K-1) + ... + binomial(N(L),L)
where N(K) > ... > N(L) >= 1, for some L. The value returned is the list [N(t) | t in 1..K] where N(t)=0 for all t < L.

The function call BinomialReprShift(N,K,up,down) computes the integer 
  binomial(N(K)  +up,    K+down) +
  binomial(N(K-1)+up,(K-1)+down) +
  ... +
  binomial(N(L)  +up,    L+down)
It is useful in generalizations of Macaulay's theorem characterizing Hilbert functions.

Example 
/**/  BinRep := BinomialRepr(13,4);
/**/  BinRep;
[1, 3, 4, 5]

/**/  BinomialReprShift(13,4,1,1);
16

See Also