cp's OEIS Frontend

Invert transform of A000009.

From Gus Wiseman, Jul 31 2022: (Start)

Also the number of ways to choose a multiset partition into distinct constant multisets of a multiset of length n that covers an initial interval of positive integers. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(4) = 11 multiset partitions are:

{{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}

{{1},{2}} {{1},{1,1}} {{1},{1,1,1}}

{{1},{2,2}} {{1,1},{2,2}}

{{2},{1,1}} {{1},{2,2,2}}

{{1},{2},{3}} {{2},{1,1,1}}

{{1},{2},{1,1}}

{{1},{2},{2,2}}

{{1},{2},{3,3}}

{{1},{3},{2,2}}

{{2},{3},{1,1}}

{{1},{2},{3},{4}}

The non-strict version is A055887.

The strongly normal non-strict version is A063834.

The strongly normal version is A270995.

(End)

A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.

a(n) = A103446(n) for n>=1; here a(0) is set to 1 in accordance with the definition and other important generating functions.

From Gus Wiseman, Dec 12 2022: (Start)

Also the number of sequences of compositions (A133494) with weakly decreasing lengths and total sum n. For example, the a(0) = 1 through a(3) = 8 sequences are:

() ((1)) ((2)) ((3))

((11)) ((12))

((1)(1)) ((21))

((111))

((1)(2))

((2)(1))

((11)(1))

((1)(1)(1))

The case of constant lengths is A101509.

The case of strictly decreasing lengths is A129519.

The case of sequences of partitions is A141199.

The case of twice-partitions is A358831.

(End)

Consider the "ordered partitions of partitions" as described in A055887. They are produced by introducing separators (a term used by J. Riordan) between the parts of a partition. If a partition has P parts, then it is possible to introduce 1, 2, ... P-1 separators. Let "|" denote such a separator. We just append 1,2,...,P-1 separators to each integer partition of n and subsequently form all permutation of the resulting list (which is composed of parts and separators).

There are some rules: If we do not append a separator, then we do not perform any permutation. Furthermore, we do not accept permutations which have a dangling separator in front of the integer parts or past them. E.g. the permutations [|,1,2,3] and [1,2,3,|] are forbidden. Furthermore, sequences of separators as "|,|" are forbidden.

Now we impose a further restriction on the permutations. Consider the elements between two separators. We call their number "occupation number". We just request that the occupation number of a ordered partition is monotonically decreasing (if we start from the left to the right of a permutation written in our notation). If we interpret a separator as a level, then we can speak of a hierarchy. E.g. we do not count [1,|,2,3,|,4] as a hierarchy, but we accept [1,2|,3,4] as a hierarchy. We thus speak of "hierarchically ordered partitions of partitions" for this sequence.

With the generating function f := z -> 1/(mul(1-z^i/mul(1-z^j,j=1..i), i=1..25)); we get the asymptotic expansion using the command equivalent (f(z),z,n);

The result is 3.788561346*exp(-n)^(-log(2)) + O(1/n*exp(-n)^(-log(2))). Let fas := n -> 3.788562346*exp(-n)^(-log(2)); then for n=60 we get fas(60)/A141199(60)= .4367915009e19/4344507472742893655 = 1.005387846.

In short, a(n) is the number of finite sequences of integer partitions with weakly decreasing lengths and total sum n. The case of twice-partitions is A358831. A version choosing compositions is A218482. The strictly decreasing case is A358836. For ordered set partitions we have A005651. For weakly decreasing bigomega see A358335. - Gus Wiseman, Dec 05 2022

a(n) is the number of finite sets of compositions with total sum n. The case of constant sums is A358904, cf. A074854. The case of distinct sums is A304961, ordered A336127. The ordered version (sequences of distinct compositions) is A358907. - Gus Wiseman, Dec 12 2022

Also the convolution triangle of A000041. - Peter Luschny, Oct 07 2022

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Labeled analog of A055887. See combstruct commands for more precise definition.

Stirling transform of A000670(n) = [1,3,13,75,...] is a(n) = [1,4,23,175,...]. - Michael Somos, Mar 04 2004

Row sums of A232598. So 2*a(n) is the number of formulas in first-order logic that have an n-place predicate, and don't include a negator. - Tilman Piesk, Nov 28 2013

An anti-run is a sequence with no adjacent equal parts.

These are partitions with no part appearing more than twice and greatest part appearing only once.

Also the number of reversed integer partitions of n whose maximal anti-runs have distinct maxima.