when this is the case. This is a generalization of the well-known result that {X} = {1} 0 (X) vanishes if (X) contains more than two nonzero parts. Thus, since the representation {X1, X2} of the 2-dimensional unimodular unitary group equals {X1 -A2}, the analysis of {l} k is of the form {k} + clI{k -2} + C2{k -4} + . . ., where cl = k -1 is the dimension of the representation (k -1, 1) of the symmetric group on k symbols, C2 = k(k -3)/2 is the dimension of the representation (k -2, 2) of this symmetric group, and so on. We do not list { m} 0 (X) when the number p of nonzero parts of (X) is m + 1, for this is the same as {m} 0 (X')where (X') = (X1-Xm+1, X2 -Xm+l ... X Xm -Xm+i) is the partition, with less than m + 1 nonzero parts, of k -(m + l)Xm+i. For example, {2} 0 13 = {2} 0 0 = {O}; {2} 0 321 = {2} 0 21; {3} 0 313 = {3} 0 2 = {6} + {2}; and so on. We conclude with the remark that the relation {m} 0 lk = {m -k + 1 } 0 k, together with the relation {m} 0 lk = {m} 0 1m,-k+l (which is an immediate consequence of the fact that {m} is of dimension m + 1), imply the relation {m} 0 k = { kI 0 m, which is known as Hermite's Law of Reciprocity. We have given previously' another proof, based on the relation ({m -I 0 k) { -1 = ({m} k -1){m -1}, of this law.I have to thank Professor E. P. Wigner for calling to my attention the importance in spectroscopy of the problem here discussed.
Proof: Here g = 111, X = 3, 0 = 10. Take 46 = 5. Clearly-X is a quadratic non-residue of 4. 3. There exists no P.D.S. of order 22 (Bruck and Ryser), i.e., no D.S. of 22 numbers (mod 463). Proof: Here g = X = 463, 0 = 22. Take 46 = 11. Clearly-X is a quadratic non-residue of 4. 4. There exists no P.D.S. of order 160. This result is new. Proof: We show that there cannot exist a D.S. of 160 numbers (mod 1592 + 159 + 1). Since 1592+ 159 + 1 0 (mod 19), we may takeX = 19. Take 4 = 3 since 0 = 159. Clearly-X is a quadratic non-residue of 46.
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