cp's OEIS Frontend

A339620 Heinz numbers of non-multigraphical partitions of even numbers.

%I A339620 #11 Feb 16 2025 08:34:01
%S A339620 3,7,10,13,19,21,22,28,29,34,37,39,43,46,52,53,55,57,61,62,66,71,76,
%T A339620 79,82,85,87,88,89,91,94,101,102,107,111,113,115,116,117,118,129,130,
%U A339620 131,133,134,136,138,139,146,148,151,155,156,159,163,166,171,172,173
%N A339620 Heinz numbers of non-multigraphical partitions of even numbers.
%C A339620 An integer partition is non-multigraphical if it does not comprise the multiset of vertex-degrees of any multigraph (multiset of non-loop edges). Multigraphical partitions are counted by A209816, non-multigraphical partitions by A000070.
%C A339620 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
%C A339620 The following are equivalent characteristics for any positive integer n:
%C A339620 (1) the multiset of prime indices of n can be partitioned into strict pairs (a multiset of edges);
%C A339620 (2) n can be factored into squarefree semiprimes;
%C A339620 (3) the unordered prime signature of n is multigraphical.
%H A339620 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphicalPartition.html">Graphical partition.</a>
%F A339620 Equals A300061 \ A320924.
%F A339620 For all n, both A181821(a(n)) and A304660(a(n)) belong to A320891.
%e A339620 The sequence of terms together with their prime indices begins:
%e A339620       3: {2}         53: {16}          94: {1,15}
%e A339620       7: {4}         55: {3,5}        101: {26}
%e A339620      10: {1,3}       57: {2,8}        102: {1,2,7}
%e A339620      13: {6}         61: {18}         107: {28}
%e A339620      19: {8}         62: {1,11}       111: {2,12}
%e A339620      21: {2,4}       66: {1,2,5}      113: {30}
%e A339620      22: {1,5}       71: {20}         115: {3,9}
%e A339620      28: {1,1,4}     76: {1,1,8}      116: {1,1,10}
%e A339620      29: {10}        79: {22}         117: {2,2,6}
%e A339620      34: {1,7}       82: {1,13}       118: {1,17}
%e A339620      37: {12}        85: {3,7}        129: {2,14}
%e A339620      39: {2,6}       87: {2,10}       130: {1,3,6}
%e A339620      43: {14}        88: {1,1,1,5}    131: {32}
%e A339620      46: {1,9}       89: {24}         133: {4,8}
%e A339620      52: {1,1,6}     91: {4,6}        134: {1,19}
%e A339620 For example, a complete lists of all loop-multigraphs with degrees (5,2,1) is:
%e A339620   {{1,1},{1,1},{1,2},{2,3}}
%e A339620   {{1,1},{1,1},{1,3},{2,2}}
%e A339620   {{1,1},{1,2},{1,2},{1,3}},
%e A339620 but since none of these is a multigraph (they have loops), the Heinz number 66 belongs to the sequence.
%t A339620 prpts[m_]:=If[Length[m]==0,{{}},Join@@Table[Prepend[#,ipr]&/@prpts[Fold[DeleteCases[#1,#2,{1},1]&,m,ipr]],{ipr,Select[Subsets[Union[m],{2}],MemberQ[#,m[[1]]]&]}]];
%t A339620 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A339620 Select[Range[100],EvenQ[Length[nrmptn[#]]]&&prpts[nrmptn[#]]=={}&]
%Y A339620 A000070 counts these partitions.
%Y A339620 A300061 is a superset.
%Y A339620 A320891 has image under A181819 equal to this set of terms.
%Y A339620 A001358 lists semiprimes, with squarefree case A006881.
%Y A339620 A002100 counts partitions into squarefree semiprimes.
%Y A339620 A320656 counts factorizations into squarefree semiprimes.
%Y A339620 The following count vertex-degree partitions and give their Heinz numbers:
%Y A339620 - A058696 counts partitions of 2n (A300061).
%Y A339620 - A000070 counts non-multigraphical partitions of 2n (A339620 [this sequence]).
%Y A339620 - A209816 counts multigraphical partitions (A320924).
%Y A339620 - A147878 counts connected multigraphical partitions (A320925).
%Y A339620 - A339655 counts non-loop-graphical partitions of 2n (A339657).
%Y A339620 - A339656 counts loop-graphical partitions (A339658).
%Y A339620 - A339617 counts non-graphical partitions of 2n (A339618).
%Y A339620 - A000569 counts graphical partitions (A320922).
%Y A339620 The following count partitions of even length and give their Heinz numbers:
%Y A339620 - A027187 has no additional conditions (A028260).
%Y A339620 - A096373 cannot be partitioned into strict pairs (A320891).
%Y A339620 - A338914 can be partitioned into strict pairs (A320911).
%Y A339620 - A338915 cannot be partitioned into distinct pairs (A320892).
%Y A339620 - A338916 can be partitioned into distinct pairs (A320912).
%Y A339620 - A339559 cannot be partitioned into distinct strict pairs (A320894).
%Y A339620 - A339560 can be partitioned into distinct strict pairs (A339561).
%Y A339620 Cf. A001055, A005117, A007717, A030229, A050320, A056239, A112798, A320655, A338899, A339113, A339661.
%K A339620 nonn
%O A339620 1,1
%A A339620 _Gus Wiseman_, Dec 18 2020