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A333941 Triangle read by rows where T(n,k) is the number of compositions of n with rotational period k.

%I A333941 #14 Jan 19 2023 22:36:23
%S A333941 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,
%T A333941 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,
%U A333941 125,120,84,32,9,0,0,2,10,45,120,210,252,210,120,45,10,0
%N A333941 Triangle read by rows where T(n,k) is the number of compositions of n with rotational period k.
%C A333941 A composition of n is a finite sequence of positive integers summing to n.
%H A333941 Andrew Howroyd, <a href="/A333941/b333941.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%F A333941 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
%e A333941 Triangle begins:
%e A333941    1
%e A333941    0   1
%e A333941    0   2   0
%e A333941    0   2   2   0
%e A333941    0   3   2   3   0
%e A333941    0   2   4   6   4   0
%e A333941    0   4   6   9   8   5   0
%e A333941    0   2   6  15  20  15   6   0
%e A333941    0   4   8  24  32  35  18   7   0
%e A333941    0   3  10  27  56  70  54  28   8   0
%e A333941    0   4  12  42  84 125 120  84  32   9   0
%e A333941    0   2  10  45 120 210 252 210 120  45  10   0
%e A333941    0   6  18  66 168 335 450 462 320 162  50  11   0
%e A333941 Row n = 6 counts the following compositions (empty columns indicated by dots):
%e A333941   .  (6)       (15)    (114)  (1113)  (11112)  .
%e A333941      (33)      (24)    (123)  (1122)  (11121)
%e A333941      (222)     (42)    (132)  (1131)  (11211)
%e A333941      (111111)  (51)    (141)  (1221)  (12111)
%e A333941                (1212)  (213)  (1311)  (21111)
%e A333941                (2121)  (231)  (2112)
%e A333941                        (312)  (2211)
%e A333941                        (321)  (3111)
%e A333941                        (411)
%t A333941 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Function[c,Length[Union[Array[RotateRight[c,#]&,Length[c]]]]==k]]],{n,0,10},{k,0,n}]
%o A333941 (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
%Y A333941 Column k = 1 is A000005.
%Y A333941 Row sums are A011782.
%Y A333941 Diagonal T(2n,n) is A045630(n).
%Y A333941 The strict version is A072574.
%Y A333941 A version counting runs is A238279.
%Y A333941 Column k = n - 1 is A254667.
%Y A333941 Aperiodic compositions are counted by A000740.
%Y A333941 Aperiodic binary words are counted by A027375.
%Y A333941 The orderless period of prime indices is A052409.
%Y A333941 Numbers whose binary expansion is periodic are A121016.
%Y A333941 Periodic compositions are counted by A178472.
%Y A333941 Period of binary expansion is A302291.
%Y A333941 Numbers whose prime signature is aperiodic are A329139.
%Y A333941 All of the following pertain to compositions in standard order (A066099):
%Y A333941 - Length is A000120.
%Y A333941 - Necklaces are A065609.
%Y A333941 - Sum is A070939.
%Y A333941 - Rotational symmetries are counted by A138904.
%Y A333941 - Constant compositions are A272919.
%Y A333941 - Lyndon compositions are A275692.
%Y A333941 - Co-Lyndon compositions are A326774.
%Y A333941 - Aperiodic compositions are A328594.
%Y A333941 - Rotational period is A333632.
%Y A333941 - Co-necklaces are A333764.
%Y A333941 - Reversed necklaces are A333943.
%Y A333941 Cf. A000031, A001037, A008965, A019536, A211100, A291166, A328595, A328596, A329312, A329313, A329326.
%K A333941 nonn,tabl
%O A333941 0,5
%A A333941 _Gus Wiseman_, Apr 16 2020