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A352489
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Weak excedance set of A122111. Numbers k <= A122111(k), where A122111 represents partition conjugation using Heinz numbers.
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13
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1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85
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OFFSET
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1,2
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COMMENTS
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The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The sequence lists all Heinz numbers of partitions whose Heinz number is less than or equal to that of their conjugate.
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LINKS
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FORMULA
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EXAMPLE
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The terms together with their prime indices begin:
1: ()
2: (1)
3: (2)
5: (3)
6: (2,1)
7: (4)
9: (2,2)
10: (3,1)
11: (5)
13: (6)
14: (4,1)
15: (3,2)
17: (7)
19: (8)
20: (3,1,1)
For example, the partition (3,2,2) has Heinz number 45 and its conjugate (3,3,1) has Heinz number 50, and 45 <= 50, so 45 is in the sequence, and 50 is not.
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Select[Range[100], #<=Times@@Prime/@conj[primeMS[#]]&]
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CROSSREFS
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These partitions are counted by A046682.
These are the positions of nonpositive terms in A352491.
A008292 is the triangle of Eulerian numbers (version without zeros).
A352522 counts compositions by weak subdiagonals, rank statistic A352515.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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