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allocated for Gus WisemanMM-numbers of lexicographically normalized multisets of multisets.

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

1,2

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 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}}

The sequence of all lexicographically normalized multisets of multisets together with their MM-numbers 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}

A subset of A320456.

MM-weight is A302242.

Non-isomorphic multiset partitions are A007716.

Cf. A056239, A112798, A317533, A330061, A330098, A330103, A330105, A330194.

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).

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Gus Wiseman, Dec 05 2019

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allocated for Gus Wiseman

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