374 MAJOR P. A. MAcMAHON ON COMBINATORIAL ANALYSIS. 



. . . ), (Xj/ijjK.j . . . ), (X^gl/jj ...),. 



successively. 



Let A,' + pi 4- V)' + . . . = A, 



the assemblage of dashed numbers being in some order identical with the assemblages 

 of undashed numbers. The new conditions include lattices enumerated by 



IV ( x iV)V 



and the totality of lattices, implied by them, is enumerated by 



the summation being for every separation of the assemblage of numbers 



*!> Ml. "! A 2 , ^ V V A 3 , /*3> "3 



into partitions 



(V/*jV )' (A'W"/ ). (Vf^V ..-)> 

 such that 



V + /*i' + "/ + = A , 



V + ^ 2 ' + "a' + = /* > 

 V + ^3' + ^s' + = " 5 



or, as it is convenient to say, for every separation of the given assemblage of numbers 

 which has the specification X, p, v . . . With this nomenclature we may say that 

 the successive row partitions have a specification p, q, r . . . and we may assert that 

 the lattices under enumeration are associated with a definite assemblage of numbers 

 and with two specifications, all three of which denote partitions of the same number, 

 SX' + 2/*' + Sv'+ . . . = X-|-/A + v+ . . . =p + q -\- r + . . . We thus asso- 

 ciate the lattice with three partitions of one number. 



There is a law of symmetry connected with these lattices the true nature of 

 which is not at once manifest ; it is not obtained by simple transposition of the above 

 lattices, and we are not permitted to simply exchange the partitions (Xfiv . . .), 

 (pqr ) preserving the assemblage of compartment numbers with the object of 

 obtaining identity of enumeration. The difficulty presents itself whenever two or 

 more partitions, (X//*/^' . . .), (A 2 'jt 2 V a ' . . .), &c. . . . are different but have the 

 same specification. 1 will obtain the true theorem by the examination of a particular 

 case. Let the assemblage of numbers be 2, 2, 1, 1, and consider the two results 



(2)9(1)2= . . . +6(2 3 )+ . . . 



