A213938 The n-th multiset representative in Abramowitz-Stegun order is a partition of a(n).

1, 2, 3, 3, 4, 6, 4, 5, 6, 7, 10, 5, 6, 7, 8, 9, 11, 15, 6, 7, 8, 9, 9, 10, 12, 12, 13, 16, 21, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, 16, 17, 18, 22, 28, 8, 9, 10, 11, 12, 11, 12, 13, 14, 15, 14, 15, 16, 17, 20, 18, 19, 21, 23, 24, 29, 36, 9, 10, 11, 12, 13, 12, 13, 14, 15, 15, 16, 18

Offset: 1

Author

Wolfdieter Lang, Jul 04 2012

Keywords

• nonn

Comments

The repetition class of a multiset is determined by a finite list of nonincreasing positive integers [e(1), e(2), ..., e(m)], which satisfy e(1) >= e(2) >= ... >= e(m) >= 1. This list will be called the multiset signature (mss). The empty multiset has as signature the empty list [] by definition. The repetition class for given signature depends on the repertoire of the m distinct objects, which will here be positive numbers. The order (cardinality) of each multiset in the repetition class is n := Sum_{j=1..m} e(j). This will be called an n-multiset with m distinct numbers. In sets the order of elements is irrelevant, and we use the convention to write a set of numbers as a list of nondecreasing numbers. As a multiset representative (msr) we chose among the members of a repetition class the one whose list entries sum to the least value. Thus the repertoire is I_m := [1, 2, ..., m] for the above given signature.

Partitions of n >= 1 (the empty partition could be included for n=0), listed in Abramowitz-Stegun (A-St) order, like on pp. 831-2 of this handbook (see A036036 for the reference, and a link to a historic paper by C. F. Hindenburg from 1779), can be used as signature if the parts are listed nonincreasingly. This is the reverse of the explicit form of a partition as given in A-St. E.g., 1^2,3 = [1,1,3] in A-St is reversed, to obtain [3,1,1] which is a signature with m=3, e(1)=3, e(2)=1 and e(3)=1. This could be called partrev(n,m,j), if the j-th partition of n with m parts, in A-St order, is labeled part(n,m,j). The empty partition could be included for n=0=m, part(0,0,1). We use partrev(n,m,j)=mss(n,m,j) for the multiset signature related to the reversed partition. The empty multiset would be the empty list [] = mss(0,0,1). In the given example the msr is then msr(5,3,1) = [1^3,2^1,3^1] = [1,1,1,2,3]. This msr is itself a partition (in the A-St version, with nondecreasing parts) of N=8. Each (nonempty) msr(n,m,j) arises thus from the partition pa(n,m,j) by reversion and 'exponentiation'. Therefore, one can carry the A-St ordering of partitions over to the msrs. This leads to an A-St type ordering of the msrs: for each increasing order n the number of distinct elements m runs from m=1,2,...,n, and lexicographic ordering is used for msrs with the same n and the same m values. This is done in the link, Table 1, for the first 194 msrs (omitting the empty multiset), corresponding to all partitions of n, for n = 1, 2, ..., 11. msr(n,m,j) (also counted as msr(k), k = 1, 2, ..., 194) is a partition of N = N(n,m,j) (or N = N(k)), given a s first entry in the 2-list [N,l] there. Note that the empty multiset is not a partition of 0, it corresponds to the empty partition. The second entry l tells that this N appears for the l-th time. E.g., msr(5,3,2)] = msr(16) = [1,1,2,2,3] belongs to the signature mss partrev(5,3,2) = [2,2,1], and is a partition of N=9, and this is the first time (l=1) that 9 appears (the next time is for k=22 or [n,m,j] = [6,2,3]. In this notation the present sequence is N=N(k), k >= 1. Here we use k-> n, N->a(n), therefore the above n, m, j should be renamed n', m', j'.

In the link, Table 2, we have given another list of msrs, counted by q = 1, 2, ..., where the A-St list of (nonempty) partitions is scanned for those that are themselves multiset representatives (when one writes the A-St partition as a list, e.g., pa(5,3,2) = [1,2,2]). Table 2 shows this list for all msrs among all partitions of N = 1, 2, ..., 20. This means that qmax=192. In A176723 we gave the characteristic array for these msrs, called there multiset representative defining partitions, and in the link there we listed the first 86 (q=1..86) of them, from N=1,...,15.

Examples

			a(1) = 1 because the first (nonempty) multiset representative (msr) is [1], a partition of 1.
a(5) = 4 because the fifth msr is [1, 1, 2] (from the fifth partition [1, 2] in A-St order and signature [2, 1]), and this is a partition of 5.
See the link for the complete Table I with a(n), n >= 1, appearing there as N(k), k >=1 .
		

Links

• Wolfdieter Lang, List of multiset representatives, k = 1..194.

Crossrefs

Cf. A176723.

Formula

The n-th representative of the repetition class of multisets defined by the signature obtained from the n-th partition of positive integers in Abramowitz-Stegun (A-St) order is a partition of a(n), n >= 1.