, satisfies: 1k C1k U1k := ) + 1 r11 ,...,r1k−1 ,s1k jr1j =n1 j=1 r1 s1k =0 r1! k−1 = ) 1j j=1 r11 ,...,r1k−1 jr1j =n1 j=1 min{n1 ,n0 +1} jr1j =n1 r1 j=1 = ) r1j =r1 j=1 k−1 ) 1j r1 =1 j=1 n1 −r1 n0 + 1 r11 min{n1 ,n0 +1} ) l k−1 1) £(k − − − = ) r1 1 n − r1 =1 =0 1 r1 n0 + 1 (−1) £ r1 where the last line follows by first noting that the inner sum of the third line is the number of compositions (ordered partitions) of n1 k into r1 1 parts, which has generating function (x + x2 − − + xk−1)r1 (Riordan 1958, p. 124). Therefore the inner sum can be generated as the · · · coefficient on xn1 in the multinomial expansion of (x + x2 + xk−1)r1 . The inner sum of binomial · · · coefficients in the fourth line corresponds to the coefficient on xn1 in the binomial expansion of of an equivalent representation of the generating function xr1 (1 xk−1)r1 /(1 x)r1 = (x + x2 + − − · · · xk−1)r1 . The coefficient in the binomial expansion must agree with the coefficient in the multinomial expansion.50 xi = k, in which case Pˆ1k (x) is also undefined, all In case 1ii, with f1k = 1 and Ln sequences that satisfy this criteria can be constructed by first forming a distinguishable permutation of the 1 runs of 1s not including the final run of 1s, which can be done in r1 k r1!/( − r1j !) ways, and second placing these runs to the left or the right of the available n0 0s, not k−1 including the r1 −1 right end point, which can be done in ( n0
1k , satisfies: U1k := ) C1k r11 ,...,r1k−1 ,s1k j=1 jr1j =n1 s1k =0 = ) r11 ,...,r1k−1 j=1 jr1j =n1 min{n1 ,n0 +1} r1! k−1 j=1 1j + 1 r1 = ) r1 =1 n0 + 1 r1 ) r11 j=1 jr1j =n1 j=1 r1j =r1 n1 −r1 k−1 j=1 1j min{n1 ,n0 +1} = ) r1 =1 n0 + 1 r1 l k−1 ) =0 r1 (−1) £ − 1 − £(k − 1) r1 − 1 where the last line follows by first noting that the inner sum of the third line is the number of compositions (ordered partitions) of n1 − k into r1 − 1 parts, which has generating function (x + x2 + · · · xk−1)r1 (Riordan 1958, p. 124). Therefore the inner sum can be generated as the coefficient on xn1 in the multinomial expansion of (x + x2 + · · · xk−1)r1 . The inner sum of binomial coefficients in the fourth line corresponds to the coefficient on xn1 in the binomial expansion of of an equivalent representation of the generating function xr1 (1 − xk−1)r1 /(1 − x)r1 = (x + x2 + · · · xk−1)r1 . The coefficient in the binomial expansion must agree with the coefficient in the multinomial expansion.50 In case 1ii, with f1k = 1 and Ln xi = k, in which case Pˆ1k (x) is also undefined, all sequences that satisfy this criteria can be constructed by first forming a distinguishable permutation of the r1 − 1 runs of 1s not including the final run of k 1s, which can be done in r1!/( k−1 r1j !) ways, and second placing these runs to the left or the right of the available n0 0s, not including the right end point, which can be done in ( n0 r1 −1 n
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