cp's OEIS Frontend

The first array is described in A126442 and is the hook case. Sequence A129305 encodes the multisets counted by A096443 and A129306.

A multiset multisystem is a finite multiset of finite multisets of positive integers. The n-th multiset multisystem is constructed by factoring n into prime numbers and then factoring each prime index into prime numbers and taking their prime indices. This produces a unique multiset multisystem for each n, and every possible multiset multisystem is so constructed as n ranges over all positive integers.

A multiset is normal if it spans an initial interval of positive integers. The type of a multiset of integers is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223.

If and only if there exists a multiset partition p whose multiset union has type h and where g = {g_1,...,g_n} is the multiset of types of the blocks of p, there exists a *combinatory separation* which is regarded as a multi-arrow p:h<=g. For example 1122<={12,11} is *not* a combinatory separation because one cannot partition a multiset of type 1122 into two blocks where one block has two distinct elements and the other block has two equal elements. Normal multisets N and combinatory separations S comprise a multi-order (N,S). The value of a(n) is the total number of *distinct* combinatory separations h<=g where h has weight n.

The term "combinatory separation" is inspired by MacMahon's inscrutable "Combinatory Analysis" (1915) which states: "A partition of any number is "separated" into "separates" by writing down a set [sic] of partitions, each partition in its own brackets, from left to right so that when all of the parts of these partitions are assembled in a single bracket, the partition separated is reproduced."

A multiset partition is a finite multiset of finite nonempty multisets of positive integers. The n-th twice-prime-factored multiset partition is constructed by factoring n into prime numbers and then factoring each prime index plus 1 into prime numbers. This produces a unique multiset of multisets of prime numbers which can then be normalized (see example) to produce each possible multiset partition as n ranges over all positive integers.

The exponents can be read off Abramowitz and Stegun, p. 831, column labeled "pi".

Here are the partitions in the order used by Abramowitz and Stegun (graded reflected colexicographic order): 0; 1; 2, 1+1; 3, 1+2, 1+1+1; 4, 1+3, 2+2, 1+1+2, 1+1+1+1; 5, 1+4, 2+3, 1+1+3, 1+2+2, 1+1+1+2, 1+1+1+1+1; ... (Cf. A036036)

Here are the partitions in graded colexicographic order: 0; 1; 2, 1+1; 3, 2+1, 1+1+1; 4, 3+1, 2+2, 2+1+1, 1+1+1+1; 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1; ... (Cf. A036037)

Since the prime signature is a partition of Omega(n), so to speak, the internal order is only a matter of convention and has no effect on the least integer with a given prime signature.

The graded colexicographic order has the advantage that the exponents are in the same order as the least integer with a given prime signature (also used on the wiki page, see links).

Embedded values include the primorial numbers 1, 2, 6, 30, 210, 2310, 30030 ... (A002110) with unordered factorizations counted by A000110 (Bell numbers) and ordered factorizations by A000670 (ordered Bell numbers).

When viewed as a table the n-th row has partition(n) (A000041(n)) terms. - Alford Arnold, Jul 31 2003

A closely related sequence, A096443(n), gives the number of partitions of the n-th multiset. - Alford Arnold, Sep 29 2005

A multiset partition is a finite multiset of finite nonempty multisets of positive integers. The n-th twice-odd-factored multiset partition is constructed by factoring 2n + 1 into prime numbers and then factoring each prime index into prime numbers and taking their prime indices.

First in a series of triangular arrays which comprise subsequences of A096443(n).

The second array begins 9 16 26 29 52 92 47 98 198 371 and when the arrays are aligned as illustrated in triangle A126441 with p(n) values they sum to A035310 which counts unordered multisets.

Let t(n, k) be the number of ways to partition the k-multiset {0,0,...,0,1,2,3,4,...,k-n} with n zeros, 0 <= n < k. Then t(n, k) = sum_i = 0..k j = 0..n S(n, j) C(i, j) p(k - n - i), where S(n, j) are Stirling numbers of the second kind, C(i, j) are the number of compositions of i distinct objects into j parts, and p is the integer partition function.

To see this, partition [n] into j blocks; there are S(n, j) partitions. For such a partition x and for each i, there are C(i, j) ways to distribute i zeros into x, because the blocks of x are all distinct. There are p(k-n-i) ways to partition the remaining k-n-i zeros. Multiplying and summing gives the result. - George Beck, Jan 10 2011

Values are also part of A096443, A129306 and A249620. Columns are also columns of the last one of these irregular triangles. See "Partitions_of_multisets" link. - Tilman Piesk, Nov 09 2014

A093936 is an analogous array for unordered factorizations.

First column is A000079. First two diagonals are A000670 and A005649.