THEORY OF THE PARTITIONS OF NUMBERS. 85 



This short demonstration suffices to establish the general relations 



nu/ \ L( oo ; a,, a,, Og, ...) 



(l)(2). 

 L(oo;m, n) 



Art. 10. Remarkable properties of the lattice functions will present themselves as 

 the investigation proceeds. A few observations may be usefully made at this point. 

 In every case the zero power of x presents itself in correspondence with that 

 permutation of the Greek letters which is in alphabetical order. 



In the case of partitions on a line the lattice is a single row of nodes ; the Greek 

 letter succession is composed entirely of the letter a and the lattice function is unity. 



A most useful property arises simply from the definition of the function, viz., 

 putting x equal to unity we find that the sum of the coefficient is 



fZ la nv^ n , (a '- a '- s+0 ' 



I J i ... ^tt m l T 1. ) (*M I , ' 



a verification of constant service. 



I seek a representation of the lattice function that shall be a constant reminder of 

 this enumerating function, and with this object in view I write the latter in the form 





_ __ 



(m.m+l . ...a! + m-l)(m-l . m . ...a a + m-2) ... U(t-s) 



...{2.3...(a w _,+ l)}(1.2.... w ) 



and I then write 



L(oo ; a,, a s , 3 , ..., a m ) 



(l)(2)...QSa) _ 

 = (m) (m+1) ... (aH-m-1). (m-1) (m) ... - 



' 



where the algebraic fraction on the dexter, which I term the outer lattice function, is 

 of fixed form, and the remaining algebraic factor IL( ; a,, a,, ..., ), which I term 

 the inner lattice function, has to be determined. 



The outer function reduces to the corresponding part of the arithmetical function 

 when x is put equal to unity; under the same circumstances the inner function 

 reduces to the sum of its own coefficients, viz., to 



U(a.-a t +t-s) -r n (t-s). 

 >,t . ' 



There is a convenience in thus postulating the expression of an outer lattice 

 function, because in every known result in regard to complete lattices the inner 



