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A357217
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Array read by descending antidiagonals: T(n,k) is the number of cycles of the permutation given by the order of elimination in the Josephus problem for n numbers and a count of k; n, k >= 1.
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2
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1, 1, 2, 1, 1, 3, 1, 2, 2, 4, 1, 1, 1, 2, 5, 1, 2, 2, 2, 1, 6, 1, 1, 1, 2, 1, 1, 7, 1, 2, 2, 2, 1, 2, 4, 8, 1, 1, 3, 2, 3, 3, 3, 2, 9, 1, 2, 2, 2, 3, 2, 2, 2, 1, 10, 1, 1, 1, 2, 1, 3, 3, 2, 3, 5, 11, 1, 2, 2, 2, 3, 2, 2, 4, 5, 2, 2, 12, 1, 1, 1, 2, 3, 1, 3, 2, 3, 1, 3, 2, 13
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OFFSET
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1,3
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COMMENTS
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n >= 2 is a Josephus_k prime if and only if T(n,k) = 1; see A163782-A163800.
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LINKS
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James Dowdy and Michael E. Mays, Josephus permutations, Journal of Combinatorial Mathematics and Combinatorial Computing 6 (1989), 125-130.
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FORMULA
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T(n,k+A003418(n)) = T(n,k), i.e., the n-th row is periodic with period A003418(n).
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EXAMPLE
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Array begins:
n\k| 1 2 3 4 5 6 7 8 9 10
---+------------------------------
1 | 1 1 1 1 1 1 1 1 1 1
2 | 2 1 2 1 2 1 2 1 2 1
3 | 3 2 1 2 1 2 3 2 1 2
4 | 4 2 2 2 2 2 2 2 2 2
5 | 5 1 1 1 3 3 1 3 3 3
6 | 6 1 2 3 2 3 2 1 2 3
7 | 7 4 3 2 3 2 3 2 5 2
8 | 8 2 2 2 4 2 2 4 6 2
9 | 9 1 3 5 3 3 3 3 3 3
10 | 10 5 2 1 2 3 2 1 2 3
For n = 4, k = 2, the order of elimination is (2,4,3,1) (row 4 of A321298). This permutation has two cycles, (1 2 4) and (3), so T(4,2) = 2.
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PROG
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(Python)
from sympy.combinatorics import Permutation
return Permutation.josephus(k, n).cycles
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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