cp's OEIS Frontend

Also the number of non-isomorphic strict multiset partitions of multiset partitions of weight n.

All sets and multisets must be finite, and only the outermost may be empty.

The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

First differs from A317145 at a(32) = 5, A317145(32) = 4.

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

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.

Also series/singleton-reduced factorizations of n with Omega(n) levels of parentheses. See A001055, A050336, A050338, A050340, etc.

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

A finite multiset is normal if it covers an initial interval of positive integers.

Consider a set N={1,2,3,...n}. We can apply the operation S~(N) on N which gives us the set partitions S~(N)=SP(N) of N. Let denote SP_i(N) such a set partition, then SP(N)={SP_1(N), SP_2(N)...,SP_B(n)} (There are B(n) set partitions of N with B(n) as the Bell number). Observe that in each SP(N) we have SP(1)={{1,2,3,...,n}} and SP(B(n))={{1},{2},{3},...,{n}} and their magnitudes are |SP(1)|=1 and |SP(B(n)|=n.

Now we perform an iteration on the set partitions SP_i(N). We set partition each SP_i(N), thus we perform S~(SP_i(N), but we exclude SP(1)={{1,2,3,...,n}} and SP(B(n))={{1},{2},{3},...,{n}} from this repetition. Otherwise an infinite recursion arises. Thus if 1 < |SP_i(N)|=m < n, then we apply S~ on SP_i(N) again and get S~(SP_i(N))= SP(SP_i(N))={SP_1(SP_i(N)),...,SP_B(m)(SP_i(N))}. We repeat this partition operation S~ on every set partition we encounter. Let denote U_k a subset of SP_i(X) were X is a set. X may be any of the subsequent set partitions. Since |U_k| < X (under the condition above on m) the repeated application of S~ will end in set partitions SP(X) with |SP(X)| = 1.

Let us consider the example N={1,2,3}. The S~(N) gives us {{1,2,3}}, {{1,2},{3}}, {{1,3},{2}}, {{2,3},{13}} and {{1},{2},{1}}. We exclude {{1,2,3}} and {{1},{2},{1}} from further partitioning. From {{1,2},{3}} we get {{{1,2},{3}}} and {{{1,2}},{{3}}}. Consider the last two partitions. They correspond to N'={1',2'} and are thus {{1',2'}} and {{1'},{2'}}. Since |{{1',2'}}|=1 and |{{1'},{2'}}|=2 these last two set partitions cannot be partitioned any further according to our condition above. In total we get {{1,2,3}}, {{1,2},{3}}, {{1,3},{2}}, {{2,3},{1}}, {{{1,2},{3}}}, {{{1,2}},{{3}}}, {{{1,3},{2}}}, {{{1,3}},{{2}}}, {{{2,3},{1}}}, {{{2,3}},{{1}}}, {{1},{2},{3}} and we have a(3)=11.

A possible application are the number of coalitions among the set N={1,2,...,n} of n persons. These persons will split into parties = subsets U_k of N. Then coalitions will form among these parties, thus we encounter sets of subsets. It is even possible that coalitions form coalitions in turn. We thus define a coalition structure as a set of repeated set partitions. For example if n=6 we could have {{1,2},{3}},{{4,5,6}}, the parties {1,2} and {3} form the coalition {{1,2},{3}}. Since {{456}}={4,5,6} one might not want to consider a single set as a coalition, but formally it is possible to do so. However, if in the example all three parties are patriotic, they may stand together in questions of national interest and the coalition structure would be {{{1,2},{3}},{{4,5,6}}}.

However, in my opinion, the usual definition of a coalition as a partition of a set falls too short.

See also A005121 = Ultradissimilarity relations on an n-set. The paper "On the Asymptotic Analysis of a Class of Linear Recurrences" (by Thomas Prellberg) outlines how to find an asymptotic formula for A005121. Perhaps this method is applicable to the present sequence as well, but one needs to have the generating function as starting point.

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. A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

A multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

A series/singleton-reduced rooted tree on a multiset m is either the multiset m itself or a sequence of series/singleton-reduced rooted trees, one on each part of a multiset partition of m that is neither minimal (all singletons) nor maximal (only one part). This is a multiset generalization of singleton-reduced phylogenetic trees (A000311).

The triangle reflects the Jordan-decomposition of the matrix of Stirling numbers of the second kind. A display of the matrix formula can be found at the Helms link which also explains the generation rule for the A()-numbers in a different way. - Gottfried Helms Apr 19 2014

From Gus Wiseman, Jan 02 2020: (Start)

Also the number of balanced reduced multisystems with atoms {1..n} and depth k. A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. For example, row n = 4 counts the following multisystems:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(End)

From Harry Richman, Mar 30 2023: (Start)

Equivalently, T(n,k) is the number of length-k chains from minimum to maximum in the lattice of set partitions of {1..n} ordered by refinement. For example, row n = 4 counts the following chains, leaving out the minimum {1|2|3|4} and maximum {1234}:

(empty) {12|3|4} {12|3|4} < {123|4}

{13|2|4} {12|3|4} < {124|3}

{14|2|3} {12|3|4} < {12|34}

{1|23|4} {13|2|4} < {123|4}

{1|24|3} {13|2|4} < {134|2}

{1|2|34} {13|2|4} < {13|24}

{123|4} {14|2|3} < {124|3}

{124|3} {14|2|3} < {134|2}

{134|2} {14|2|3} < {14|23}

{1|234} {1|23|4} < {123|4}

{12|34} {1|23|4} < {1|234}

{13|24} {1|23|4} < {14|23}

{14|23} {1|24|3} < {124|3}

{1|24|3} < {1|234}

{1|24|3} < {13|24}

{1|2|34} < {134|2}

{1|2|34} < {1|234}

{1|2|34} < {12|34}

(End)

Also the number of cells of dimension k in the fine subdivision of the Bergman complex of the complete graph on n vertices. - Harry Richman, Mar 30 2023

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

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. A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.