In connection with string theory I encountered the following problem:

Given the set M_N of all semi-standard Young tableaux of size N (i.e. all fillings of Ferrers diagrams with natural numbers with weakly increasing rows and strictly increasing columns, the content of which sum to N).

The conjecture, which I cannot prove and for which I know no counterexample states:

M_N is the union of disjoint families, where each family consists of a father diagram and all his daughters, which are obtained from the father diagram by deletion of one box.

The conjecture holds for N<=8

http://www.itp.uni-hannover.de/~dragon/young.ps or

http://www.itp.uni-hannover.de/~dragon/young.pdf

There numbers a,b are attached to the Ferrers diagrams, where a is the number of semi-standard Young Tableaux of size N and b the number of fathers, who remain after the daughters have filled up their families.

Light cone string theory proves that the conjecture holds for all M_N with less than 25 rows, i.e. for N < 325. But string theory provides no counterexample for N >= 325.

Any suggestions, e.g. how to compute the number a(N,lambda) of Young diagrams with size N and shape lambda, would be helpful.

Norbert Dragon

-- http://www.itp.uni-hannover.de/~dragon/

Superstition brings bad luck.