W. M. Y. Goh and E. Schmutz. The number of distinct part sizes in a random integer partitions. Journal of Combinatorial Theory, series A 69 (1995) 149--158.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....components in general (decomposable) combinatorial structures. In particular, he rederived the result of Erdos and Lehner for the mean number of distinct summands in a random partition. Central limit theorems for the number of distinct summands in partitions were derived by Goh and Schmutz [21]; see also Schmutz [45] Local limit theorems were studied by Hwang [26] The corresponding problems for compositions are, unlike most other ones, more complicated and first treated by Knopfmacher and Mays [31, 32] They derived some combinatorial properties of the number and sum of distinct parts ....

.... 1) 3n) ff 1) 2 (ff Gamma1) 2 ; if ff Gamma1: 4. 3 Partitions with small number of distinct summands Central and local limit theorems for the number of distinct parts in a random partition have been studied in Hwang [26] under a scheme essentially due to Meinardus [38] see also [21, 45]. We consider in this section the distribution of integer partitions with small number of distinct parts. Our arguments rely on some new combinatorial inequalities which are interesting per se. Define two quantities by the generating functions: q (n; m)u 1 uz (n; ....

W. M. Y. Goh and E. Schmutz. The number of distinct part sizes in a random integer partitions. Journal of Combinatorial Theory, series A 69 (1995) 149--158.

.... (n) in the theory of primes (cf. 38] this problem is rarely discussed in the theory of partitions. It was first briefly mentioned in [7] in the case = Z . Wilf [40] introduced the study of distinct components (or sizes of components) in general combinatorial structures. Then Goh and Schmutz [14] derived a central limit theorem for the number of summands for = Z . The latter result was then extended by Schmutz [34] to multivariate cases under Meinardus s scheme. We shall further extend their results by establishing the corresponding local limit theorem (in univariate case) under weaker ....

W. M. Y. Goh and E. Schmutz, The number of distinct part sizes in a random integer partitions, Journal of Combinatorial Theory, series A, 69 (1995), 149--158.

....in the theory of primes (see [38] this problem is rarely discussed in the theory of partitions. It was first briefly mentioned in [7] in the case # = Z . Wilf [40] introduced the study of distinct components (or sizes of components) in general combinatorial structures. Then Goh and Schmutz [14] derived a central limit theorem for the number of summands for # = Z . The latter result was then extended by Schmutz [34] to multivariate cases under Meinardus s scheme. We further improve and extend their results by establishing the corresponding local limit theorem (in univariate case) under ....

W. M. Y. Goh and E. Schmutz, The number of distinct part sizes in a random integer partitions, Journal of Combinatorial Theory, series A, 69 (1995), 149--158.

....components in general (decomposable) combinatorial structures. In particular, he rederived the result of Erdos and Lehner for the mean number of distinct summands in a random partition. Central limit theorems for the number of distinct summands in partitions were derived by Goh and Schmutz [21]; see also Schmutz [45] Local limit theorems were studied by Hwang [26] The corresponding problems for compositions are, unlike most other ones, more complicated and first treated by Knopfmacher and Mays [31, 32] They derived some combinatorial properties of the number and sum of distinct parts ....

....# # # P (#) if # 2#(# 1) 3n) # 1) 2 1. 4. 3 Partitions with small number of distinct summands Central and local limit theorems for the number of distinct parts in a random # partition have been studied in Hwang [26] under a scheme essentially due to Meinardus [38] see also [21, 45]. We consider in this section the distribution of integer partitions with small number of distinct parts. Our arguments rely on some new combinatorial inequalities which are interesting per se. Define two quantities by the generating functions: q # (n, m)u 1 uz ## (n, ....

W. M. Y. Goh and E. Schmutz. The number of distinct part sizes in a random integer partitions. Journal of Combinatorial Theory, series A 69 (1995) 149--158.

....is, X j ( j) let I j be the indicator of the event fX j 1g, and let I j;m be the indicator of the event fX j = mg; j 1. Then D n = P j1 I j is the total number of different part sizes in the random partition. Of course, E D n = hffii n p 6n= see Sections 2 and 3. Goh and Schmutz [6] proved the asymptotic normality of D n , from which it follows that D n is asymptotic to p 6n= in probability as well. Likewise, D n;m = P j1 I j;m is the total number of part sizes of multiplicity m, and ED n;m = hffi m i n 7 is sharply estimated in Section 4. We see that, given the ....

....n . So, intuitively, one is justified in replacing D n in (15) by n 1 2 c Gamma1 . To do this rigorously though, we need to know how unlikely is the event A n : aefi fi fi fi D n n 1 2 c Gamma1 Gamma 1 fi fi fi fi oe ; Such an estimate does not follow from the results in [3] [6], and [15] Instead, we get a good bound by using the conditioning device, suggested for the integer partitions by Fristedt [4] see also [11] and patterned after the analogous treatment of random permutations by Shepp and Lloyd [13] Namely, introduce the sequence of independent random ....

Goh, W. M. Y. and Schmutz, E., The number of distinct part sizes in a random integer partition, J. Combin. Theory Ser. A 69 (1995), 149-158.

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