### THE COMPLETE AMALGAMATION INTO BLOCKS, BY WEIGHTED MEANS, OF A FINITE SET OF REAL NUMBERS

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The problem discussed below arose in Bartholomew (1959 a), in which is proposed a test of homogeneity of 'ordered' alternatives. On p. 40 of this paper Bartholomew defined the probability P(l, k; al, a2, ..., ak) and proceeded to determine its values for 1 < 1 < k < 4 and general ai (i 1, ..., k). In the special, but important, case of equal weights (a, = a2 = ... = ak) it may readily be verified that P(l, k; al, a2, ..., ak) is independent of the common value of the ai, and thus may be written P(l, k), say; this notation departs slightly from Bartholomew's in the use of a circumflex, but this is done to avoid confusion in the Appendix. Bartholomew determined values of P(l, k) for 1 < 1 < k < 5 and, on the basis of these results, conjectured on p. 43 the recurrence relations

[1] D. J. Bartholomew,et al. A TEST OF HOMOGENEITY FOR ORDERED ALTERNATIVES. II , 1959 .

[2] William Feller,et al. An Introduction to Probability Theory and Its Applications , 1951 .

[3] John Riordan,et al. Introduction to Combinatorial Analysis , 1958 .