|
| |
|
|
A330061
|
|
MM-number of the VDD-normalization of the multiset of multisets with MM-number n.
|
|
14
|
|
|
|
1, 2, 3, 4, 3, 6, 7, 8, 9, 6, 3, 12, 13, 14, 15, 16, 3, 18, 19, 12, 21, 6, 7, 24, 9, 26, 27, 28, 13, 30, 3, 32, 15, 6, 35, 36, 37, 38, 39, 24, 3, 42, 13, 12, 45, 14, 13, 48, 49, 18, 15, 52, 53, 54, 15, 56, 57, 26, 3, 60, 37, 6, 63, 64, 39, 30, 3, 12, 35, 70
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
We define the VDD (vertex-degrees decreasing) normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, then selecting only the representatives whose vertex-degrees are weakly decreasing, and finally taking the least of these representatives, where the ordering of multisets is first by length and then lexicographically.
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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
Brute-force: 43287: {{1},{2,3},{2,2,4}}
Lexicographic: 43143: {{1},{2,4},{2,2,3}}
VDD: 15515: {{2},{1,3},{1,1,4}}
MM: 15265: {{2},{1,4},{1,1,3}}
|
|
|
LINKS
|
|
|
|
MATHEMATICA
|
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
sysnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]], sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]], i}, {i, Length[Union@@m]}]], First[Sort[sysnorm[m, 1]]]];
sysnorm[m_, aft_]:=If[Length[Union@@m]<=aft, {m}, With[{mx=Table[Count[m, i, {2}], {i, Select[Union@@m, #>=aft&]}]}, Union@@(sysnorm[#, aft+1]&/@Union[Table[Map[Sort, m/.{par+aft-1->aft, aft->par+aft-1}, {0, 1}], {par, First/@Position[mx, Max[mx]]}]])]];
Table[Map[Times@@Prime/@#&, sysnorm[primeMS/@primeMS[n]], {0, 1}], {n, 100}]
|
|
|
CROSSREFS
|
This sequence is idempotent and its image/fixed points are A330060.
Non-isomorphic multiset partitions are A007716.
Cf. A000612, A055621, A056239, A112798, A283877, A316983, A317533, A320456, A330098, A330102, A330103, A330105.
Other fixed points:
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
