A340651 - OEIS

%I #13 Jan 15 2024 20:29:17

%S 1,1,2,4,11,26,77,220,677,2098,6756,22101,74264,253684,883795,3130432,

%T 11275246,41240180,153117873,576634463,2201600769,8517634249,

%U 33378499157,132438117118,531873247805,2161293783123,8883906870289,36928576428885,155196725172548,659272353608609,2830200765183775

%N Number of non-isomorphic cross-balanced multiset partitions of weight n.

%C We define a multiset partition to be cross-balanced if it uses exactly as many distinct vertices as the greatest size of a part.

%H Andrew Howroyd, <a href="/A340651/b340651.txt">Table of n, a(n) for n = 0..50</a>

%e Non-isomorphic representatives of the a(1) = 1 through a(5) = 16 multiset partitions:

%e   {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}

%e          {{1},{1}}  {{1},{2,2}}    {{1,1},{2,2}}

%e                     {{2},{1,2}}    {{1,2},{1,2}}

%e                     {{1},{1},{1}}  {{1,2},{2,2}}

%e                                    {{1},{2,3,3}}

%e                                    {{3},{1,2,3}}

%e                                    {{1},{1},{2,2}}

%e                                    {{1},{2},{1,2}}

%e                                    {{1},{2},{2,2}}

%e                                    {{2},{2},{1,2}}

%e                                    {{1},{1},{1},{1}}

%o (PARI) \\ See A340652 for G.

%o seq(n)={Vec(1 + sum(k=1,n, G(k,n,k) - G(k-1,n,k) - G(k,n,k-1) + G(k-1,n,k-1)))} \\ _Andrew Howroyd_, Jan 15 2024

%Y The co-balanced version is A319616.

%Y The balanced version is A340600.

%Y The twice-balanced version is A340652.

%Y The version for factorizations is A340654.

%Y A007716 counts non-isomorphic multiset partitions.

%Y A007718 counts non-isomorphic connected multiset partitions.

%Y A316980 counts non-isomorphic strict multiset partitions.

%Y Other balance-related sequences:

%Y - A047993 counts balanced partitions.

%Y - A106529 lists balanced numbers.

%Y - A340596 counts co-balanced factorizations.

%Y - A340599 counts alt-balanced factorizations.

%Y - A340600 counts unlabeled balanced multiset partitions.

%Y - A340653 counts balanced factorizations.

%Y Cf. A001055, A007717, A064174, A316983, A320663, A324522, A339888, A340655.

%K nonn

%O 0,3

%A _Gus Wiseman_, Feb 05 2021

%E a(11) onwards from _Andrew Howroyd_, Jan 15 2024