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A330121
MM-numbers of lexicographically normalized multiset partitions.
19
1, 3, 7, 9, 13, 15, 19, 21, 27, 37, 39, 45, 49, 53, 57, 63, 69, 81, 89, 91, 105, 111, 113, 117, 131, 133, 135, 141, 147, 151, 159, 161, 165, 169, 171, 183, 189, 195, 207, 223, 225, 243, 247, 259, 267, 273, 281, 285, 309, 311, 315, 329, 333, 339, 343, 351, 359
OFFSET
1,2
COMMENTS
First differs from A330107 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 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}}
EXAMPLE
The sequence of all lexicographically normalized multiset partitions together with their MM-numbers begins:
1: 63: {1}{1}{11} 159: {1}{1111}
3: {1} 69: {1}{22} 161: {11}{22}
7: {11} 81: {1}{1}{1}{1} 165: {1}{2}{3}
9: {1}{1} 89: {1112} 169: {12}{12}
13: {12} 91: {11}{12} 171: {1}{1}{111}
15: {1}{2} 105: {1}{2}{11} 183: {1}{122}
19: {111} 111: {1}{112} 189: {1}{1}{1}{11}
21: {1}{11} 113: {123} 195: {1}{2}{12}
27: {1}{1}{1} 117: {1}{1}{12} 207: {1}{1}{22}
37: {112} 131: {11111} 223: {11112}
39: {1}{12} 133: {11}{111} 225: {1}{1}{2}{2}
45: {1}{1}{2} 135: {1}{1}{1}{2} 243: {1}{1}{1}{1}{1}
49: {11}{11} 141: {1}{23} 247: {12}{111}
53: {1111} 147: {1}{11}{11} 259: {11}{112}
57: {1}{111} 151: {1122} 267: {1}{1112}
CROSSREFS
Equals the odd terms of A330120.
A subset of A320634.
MM-weight is A302242.
Non-isomorphic multiset partitions are A007716.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).
Sequence in context: A330122 A330097 A330107 * A111225 A032678 A073671
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 05 2019
STATUS
approved