TIIKORY OF THE PARTITIONS OP NUMBERS. 95 



Since n m and m" are conjugate partitions, we may take n^m without loss of 

 generality. The number of dividing lines is 



Hence we have sub-lattice functions of all orders from zero to (nl)(ml). It 

 will be observed that the permutation, above written, possesses symmetry in that it 

 is unchanged by writing a m _ J+1 for a, and inverting the order. 



The same method is applicable to the determination of the maximum number of 

 dividing lines appertaining to permutations derived from an incomplete lattice. Thus, 

 if the letters be ai 3 a a a a 3 , 



a, a, a, 



8 



the reading parallel to the arrow gives 



Ott, Oty Oti OL*t Oto OL\ t 



The highest order of sub-lattice functions when the letters are 



will be found to have higher and lower limits 2n n t and 2n n, m+1 respectively, 

 the actual value depending upon the magnitudes of n it n,, ..., n m . The lower limit is 

 the actual value when the lattice is complete. 



Art. 19. The next point is the determination of the expression of L ( ,_ 1)(m _, ) ( oo; n m ), 

 or of L( B _ 1)(m _ 1) ( oo, m, n) as it may be also written. 



The dividing lines occur in groups 



(i) In m 2 groups, containing 1, 2, ..., m 2 lines respectively ; 

 (ii) In n m+l groups, each containing m 1 lines; 

 (iii) In m 2 groups, containing m 2, m 1, ..., 2, 1 lines respectively. 



Let the exponent of x sought be 71-1 + 7^ + 773 ; TT,, ir 3> ir 3 , corresponding to (i), (ii), and 

 (iii), respectively. 



.2 a + 2.3 a +3.4 a +... tom-2 terms), 

 i(m-2) a (m-l) a +i(m-2Hm-l)(2m-3)+i(m-2)(m-l), 



i(m-l)m a +(rn-l)m(m+2) + (wi-l)w( + 4)+... ton-m+1 terms, 

 %(m l)mn(n m+ 1). 



-7 + mn-8+wn-9)+... tom-2 terms, 



