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A213938
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The n-th multiset representative in Abramowitz-Stegun order is a partition of a(n).
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4
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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 .
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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