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Totally co-strong partitions are counted by A332275.

Totally co-strong partitions are counted by A332275.

The dual (co-strong) version is A317257.

The normal version appears to be 1, 2.

The sequence of terms together with their prime indices begins:

1: {} 17: {7} 35: {3,4}

2: {1} 18: {1,2,2} 36: {1,1,2,2}

3: {2} 19: {8} 37: {12}

4: {1,1} 21: {2,4} 38: {1,8}

5: {3} 22: {1,5} 39: {2,6}

6: {1,2} 23: {9} 41: {13}

7: {4} 25: {3,3} 42: {1,2,4}

8: {1,1,1} 26: {1,6} 43: {14}

9: {2,2} 27: {2,2,2} 46: {1,9}

10: {1,3} 29: {10} 47: {15}

11: {5} 30: {1,2,3} 49: {4,4}

13: {6} 31: {11} 50: {1,3,3}

14: {1,4} 32: {1,1,1,1,1} 51: {2,7}

15: {2,3} 33: {2,5} 53: {16}

16: {1,1,1,1} 34: {1,7} 54: {1,2,2,2}

allocated for Gus WisemanHeinz numbers of alternately strong integer partitions.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83

1,2

First differs from A304678 in lacking 450.

First differs from A316529 (the totally strong version) in having 150.

A sequence is alternately strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and, when reversed, are themselves an alternately strong sequence.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

The sequence of terms together with their prime indices begins:

1: {} 17: {7} 35: {3,4}

2: {1} 18: {1,2,2} 36: {1,1,2,2}

3: {2} 19: {8} 37: {12}

4: {1,1} 21: {2,4} 38: {1,8}

5: {3} 22: {1,5} 39: {2,6}

6: {1,2} 23: {9} 41: {13}

7: {4} 25: {3,3} 42: {1,2,4}

8: {1,1,1} 26: {1,6} 43: {14}

9: {2,2} 27: {2,2,2} 46: {1,9}

10: {1,3} 29: {10} 47: {15}

11: {5} 30: {1,2,3} 49: {4,4}

13: {6} 31: {11} 50: {1,3,3}

14: {1,4} 32: {1,1,1,1,1} 51: {2,7}

15: {2,3} 33: {2,5} 53: {16}

16: {1,1,1,1} 34: {1,7} 54: {1,2,2,2}

The sequence does not contain 450, the Heinz number of (3,3,2,2,1), because, while the multiplicities are weakly decreasing, their reverse (1,2,2) does not have weakly decreasing multiplicities.

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

altstrQ[q_]:=Or[q=={}, q=={1}, And[GreaterEqual@@Length/@Split[q], altstrQ[Reverse[Length/@Split[q]]]]];

Select[Range[100], altstrQ[Reverse[primeMS[#]]]&]

The normal version appears to be 1, 2.

The dual (co-strong) version is A317257.

The case of reversed partitions is (also) A317257.

The total version is A316529.

Totally co-strong partitions are counted by A332275.

These partitions are counted by A332339.

Alternately co-strong compositions are counted by A332338.

Cf. A000041, A100883, A181819, A182850, A182857, A304660, A305563, A316496, A317256, A332292, A332340.

allocated

nonn

Gus Wiseman, Jun 09 2020

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editing

allocated for Gus Wiseman

allocated

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