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A261765
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Triangle read by rows: T(n,k) is the number of subpermutations of an n-set, whose orbits are each of size at most k with at least one orbit of size exactly k, and without fixed points. Equivalently, T(n,k) is the number of partial derangements of an n-set each of whose orbits is of size at most k with at least one orbit of size exactly k, and without fixed points.
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5
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1, 1, 0, 1, 0, 3, 1, 0, 9, 8, 1, 0, 45, 32, 30, 1, 0, 165, 320, 150, 144, 1, 0, 855, 2240, 1800, 864, 840, 1, 0, 3843, 17360, 18900, 12096, 5880, 5760, 1, 0, 21819, 146048, 195300, 145152, 94080, 46080, 45360, 1, 0, 114075, 1256192, 2120580, 1959552, 1270080, 829440, 408240, 403200
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OFFSET
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0,6
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COMMENTS
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REFERENCES
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A. Laradji and A. Umar, On the number of subpermutations with fixed orbit size, Ars Combinatoria, 109 (2013), 447-460.
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LINKS
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FORMULA
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EXAMPLE
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T(n,1) = 0 because there is no (partial) derangement with an orbit of size 1.
T(3,2) = 9 because there are 9 subpermutations on {1,2,3}, whose orbits are each of size at most 2 with at least one orbit of size exactly 2, and without fixed points, namely: (1 2 --> 2 1), (1 3 --> 3 1), (2 3 --> 3 2), (1-->2), (1-->3), (2-->1), (2-->3), (3-->1), (3-->2).
Triangle starts:
1;
1, 0;
1, 0, 3;
1, 0, 9, 8;
1, 0, 45, 32, 30;
1, 0, 165, 320, 150, 144;
1, 0, 855, 2240, 1800, 864, 840;
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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