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A333941
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Triangle read by rows where T(n,k) is the number of compositions of n with rotational period k.
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6
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1, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 3, 2, 3, 0, 0, 2, 4, 6, 4, 0, 0, 4, 6, 9, 8, 5, 0, 0, 2, 6, 15, 20, 15, 6, 0, 0, 4, 8, 24, 32, 35, 18, 7, 0, 0, 3, 10, 27, 56, 70, 54, 28, 8, 0, 0, 4, 12, 42, 84, 125, 120, 84, 32, 9, 0, 0, 2, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0
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OFFSET
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0,5
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COMMENTS
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A composition of n is a finite sequence of positive integers summing to n.
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LINKS
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FORMULA
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T(n,k) = Sum_{m|n} Sum_{d|gcd(k,m)} mu(d)*binomial(m/d-1, k/d-1) for n > 0. - Andrew Howroyd, Jan 19 2023
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EXAMPLE
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Triangle begins:
1
0 1
0 2 0
0 2 2 0
0 3 2 3 0
0 2 4 6 4 0
0 4 6 9 8 5 0
0 2 6 15 20 15 6 0
0 4 8 24 32 35 18 7 0
0 3 10 27 56 70 54 28 8 0
0 4 12 42 84 125 120 84 32 9 0
0 2 10 45 120 210 252 210 120 45 10 0
0 6 18 66 168 335 450 462 320 162 50 11 0
Row n = 6 counts the following compositions (empty columns indicated by dots):
. (6) (15) (114) (1113) (11112) .
(33) (24) (123) (1122) (11121)
(222) (42) (132) (1131) (11211)
(111111) (51) (141) (1221) (12111)
(1212) (213) (1311) (21111)
(2121) (231) (2112)
(312) (2211)
(321) (3111)
(411)
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MATHEMATICA
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Function[c, Length[Union[Array[RotateRight[c, #]&, Length[c]]]]==k]]], {n, 0, 10}, {k, 0, n}]
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PROG
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(PARI) T(n, k)=if(n==0, k==0, sumdiv(n, m, sumdiv(gcd(k, m), d, moebius(d)*binomial(m/d-1, k/d-1)))) \\ Andrew Howroyd, Jan 19 2023
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CROSSREFS
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A version counting runs is A238279.
Aperiodic compositions are counted by A000740.
Aperiodic binary words are counted by A027375.
The orderless period of prime indices is A052409.
Numbers whose binary expansion is periodic are A121016.
Periodic compositions are counted by A178472.
Period of binary expansion is A302291.
Numbers whose prime signature is aperiodic are A329139.
All of the following pertain to compositions in standard order (A066099):
- Rotational symmetries are counted by A138904.
- Constant compositions are A272919.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
Cf. A000031, A001037, A008965, A019536, A211100, A291166, A328595, A328596, A329312, A329313, A329326.
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KEYWORD
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AUTHOR
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STATUS
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approved
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