Partitions and Their Lattices

Ferrers graphs and tables of partitions are treated as vectors. Matrix operations are used for simple proofs of identities concerning partitions. Interpreting partitions as vectors gives a possibility to generalize partitions on negative numbers. Partitions are then tabulated into lattices and some properties of these lattices are studied. There appears a new identity counting isoscele Ferrers graphs. The lattices form the base for tabulating combinatorial identities. Partitions of a natural number m into n parts were introduced into mathematics by Euler. The analytical formula for finding the number of partitions was derived by Ramanudjan and Hardy [1]. Ramanudjan was a mathematical genius from India. He was sure that it was possible to calculate the number of partitions exactly for any number m. He found the solution in cooperation with his tutor, the English mathematician Hardy. It is rather complicated formula derived by higher mathematical techniques. We will use only simple recursive methods for different relations between partitions. Steve Weinberg in his lecture [2] about importance of mathematics for physics mentioned that partitions got importance for theoretical physics, even if Hardy did not want to study practical problems. But partitions were used in physics before Hardy by Boltzmann [3]. He used this notion for splitting m quanta of energy between n particles in connection with his notion of entropy. He called partitions complexions, considering them to be orbits in phase space. His idea was forgoten.