

A330120


MMnumbers of lexicographically normalized multisets of multisets.


19



1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 21, 24, 26, 27, 28, 30, 32, 36, 37, 38, 39, 42, 45, 48, 49, 52, 53, 54, 56, 57, 60, 63, 64, 69, 72, 74, 76, 78, 81, 84, 89, 90, 91, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117, 120, 126, 128, 131, 133
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OFFSET

1,2


COMMENTS

First differs from A330104 in lacking 435 and having 429, with corresponding multisets of multisets 435: {{1},{2},{1,3}} and 429: {{1},{3},{1,2}}.
We define the lexicographic 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, and finally taking the lexicographically least of these representatives.
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 MMnumber 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 MMnumber 78 is {{},{1},{1,2}}.
For example, 15301 is the MMnumber of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MMnumbers:
Bruteforce: 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

Table of n, a(n) for n=1..64.


EXAMPLE

The sequence of all lexicographically normalized multisets of multisets together with their MMnumbers begins:
1: 0 21: {1}{11} 52: {}{}{12} 89: {1112}
2: {} 24: {}{}{}{1} 53: {1111} 90: {}{1}{1}{2}
3: {1} 26: {}{12} 54: {}{1}{1}{1} 91: {11}{12}
4: {}{} 27: {1}{1}{1} 56: {}{}{}{11} 96: {}{}{}{}{}{1}
6: {}{1} 28: {}{}{11} 57: {1}{111} 98: {}{11}{11}
7: {11} 30: {}{1}{2} 60: {}{}{1}{2} 104: {}{}{}{12}
8: {}{}{} 32: {}{}{}{}{} 63: {1}{1}{11} 105: {1}{2}{11}
9: {1}{1} 36: {}{}{1}{1} 64: {}{}{}{}{}{} 106: {}{1111}
12: {}{}{1} 37: {112} 69: {1}{22} 108: {}{}{1}{1}{1}
13: {12} 38: {}{111} 72: {}{}{}{1}{1} 111: {1}{112}
14: {}{11} 39: {1}{12} 74: {}{112} 112: {}{}{}{}{11}
15: {1}{2} 42: {}{1}{11} 76: {}{}{111} 113: {123}
16: {}{}{}{} 45: {1}{1}{2} 78: {}{1}{12} 114: {}{1}{111}
18: {}{1}{1} 48: {}{}{}{}{1} 81: {1}{1}{1}{1} 117: {1}{1}{12}
19: {111} 49: {11}{11} 84: {}{}{1}{11} 120: {}{}{}{1}{2}


CROSSREFS

A subset of A320456.
MMweight is A302242.
Nonisomorphic multiset partitions are A007716.
Cf. A056239, A112798, A317533, A330061, A330098, A330103, A330105, A330194.
Other fixed points:
 Bruteforce: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (setsystems).
 Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (setsystems).
 VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (setsystems).
 MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (setsystems).
 BII: A330109 (setsystems).
Sequence in context: A330060 A330108 A330104 * A239015 A030706 A285986
Adjacent sequences: A330117 A330118 A330119 * A330121 A330122 A330123


KEYWORD

nonn


AUTHOR

Gus Wiseman, Dec 05 2019


STATUS

approved



