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approved

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proposed

These partitions are counted by A209816.

The conjugate version is A344413, counted by A209816.

A209816 counts multigraphical partitions.

Cf. A000041, A000070, A000569, A007717, A096373, A265640, A283877, A306005, A318361, A320459, A320911, A320922, A320923, A320925.

From Gus Wiseman, May 23 2021: (Start)

The sequence of terms together with their prime indices and a multigraph realizing each begins:

1: () | {}

4: (11) | {{1,2}}

9: (22) | {{1,2},{1,2}}

12: (112) | {{1,3},{2,3}}

16: (1111) | {{1,2},{3,4}}

25: (33) | {{1,2},{1,2},{1,2}}

27: (222) | {{1,2},{1,3},{2,3}}

30: (123) | {{1,3},{2,3},{2,3}}

36: (1122) | {{1,2},{3,4},{3,4}}

40: (1113) | {{1,4},{2,4},{3,4}}

48: (11112) | {{1,2},{3,5},{4,5}}

49: (44) | {{1,2},{1,2},{1,2},{1,2}}

63: (224) | {{1,3},{1,3},{2,3},{2,3}}

(End)

This sequence The case with odd weights is A320924, counted by A209816A322109.

Allowing odd weights gives A322109, counted by A110618The conjugate case of equality is A340387.

The conjugate case of equality version with odd weights allowed is A340387, counted by A035363A344291.

The conjugate opposite version with odd weights allowed is A344291, counted by A110618A344292.

The conjugate opposite version with odd weights allowed is A344292, counted by A000070A344296.

The opposite version with odd weights allowed is A344296, counted by A025065.

The conjugate opposite version with odd weights allowed is A344414, counted by A025065.

The case of equality is A344415, counted by A035363.

The opposite version is A344416, counted by A000070.

A000070 counts non-multigraphical partitions.

A025065 counts palindromic partitions.

A035363 counts partitions into even parts.

A110618 counts partitions that are the vertex-degrees of some set multipartition with no singletons.

A209816 counts multigraphical partitions.

Cf. A000041, A000070, A000569, A007717, A096373, A265640, A283877, A306005, A318361, A320459, A320911, A320922, A320923, A320925.

Also Heinz numbers of integer partitions of even numbers whose greatest part is less than or equal to half the sum of parts, i.e., numbers n whose sum of prime indices A056239(n) is even and at least twice the greatest prime index A061395(n). - Gus Wiseman, May 23 2021

Members m of A300061 such that A061395(m) <= A056239(m)/2. - Gus Wiseman, May 23 2021

This sequence is A320924, counted by A209816.

Allowing odd weights gives A322109, counted by A110618.

The conjugate case of equality is A340387, counted by A035363.

The conjugate version with odd weights allowed is A344291, counted by A110618.

The conjugate opposite version is A344292, counted by A000070.

The opposite version with odd weights allowed is A344296, counted by A025065.

The conjugate version is A344413, counted by A209816.

The conjugate opposite version with odd weights allowed is A344414, counted by A025065.

The case of equality is A344415, counted by A035363.

The opposite version is A344416, counted by A000070.

A056239 adds up prime indices, row sums of A112798.

A334201 adds up all prime indices except the greatest.

Cf. A000070, A000041, A000569, A007717, A025065, A056239, A096373, A112798, A209816, A300061, A265640, A283877, A306005, A318361, A320459, A320911, A320922, A320923, A320925.

approved

editing

proposed

approved

editing

proposed

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). An integer partition is multigraphical if it comprises the multiset of vertex-degrees of some multigraph.

An integer partition is multigraphical if it comprises the multiset of vertex-degrees of some multigraph.

Cf. A000070, A000569, A007717, A025065, A056239, A096373, A112798, A209816, A300061, A320459, A320911, A320922, A320925.