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Triangle read by rows: T(n,k) is the number of compositions of n into k parts x_1, x_2, ..., x_k such that gcd(x_i, x_j) = 1 for all i != j (where 1 <= k <= n).
3

%I #29 Nov 12 2020 22:18:36

%S 1,1,1,1,2,1,1,2,3,1,1,4,3,4,1,1,2,9,4,5,1,1,6,3,16,5,6,1,1,4,15,4,25,

%T 6,7,1,1,6,9,28,5,36,7,8,1,1,4,21,16,45,6,49,8,9,1,1,10,9,52,25,66,7,

%U 64,9,10,1,1,4,39,16,105,36,91,8,81,10,11,1,1,12,9,100,25,186,49,120,9,100,11,12,1,1,6,45,16,205,36,301,64,153,10,121,12,13,1

%N Triangle read by rows: T(n,k) is the number of compositions of n into k parts x_1, x_2, ..., x_k such that gcd(x_i, x_j) = 1 for all i != j (where 1 <= k <= n).

%C See A101391 for the triangle T(n,k) = number of compositions of n into k parts x_1, x_2, ..., x_k such that gcd(x_1,x_2,...,x_k) = 1 (2 <= k <= n).

%H Alois P. Heinz, <a href="/A282748/b282748.txt">Rows n = 1..200, flattened</a> (first 100 rows from Chai Wah Wu)

%H Temba Shonhiwa, <a href="http://www.fq.math.ca/Papers1/44-4/quarttemba04_2006.pdf">Compositions with pairwise relatively prime summands within a restricted setting</a>, Fibonacci Quart. 44 (2006), no. 4, 316-323.

%F It seems that no general formula or recurrence is known, although Shonhiwa gives formulas for a few of the early diagonals.

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 2, 3, 1;

%e 1, 4, 3, 4, 1;

%e 1, 2, 9, 4, 5, 1;

%e 1, 6, 3, 16, 5, 6, 1;

%e 1, 4, 15, 4, 25, 6, 7, 1;

%e 1, 6, 9, 28, 5, 36, 7, 8, 1;

%e 1, 4, 21, 16, 45, 6, 49, 8, 9, 1;

%e 1, 10, 9, 52, 25, 66, 7, 64, 9, 10, 1;

%e 1, 4, 39, 16, 105, 36, 91, 8, 81, 10, 11, 1;

%e 1, 12, 9, 100, 25, 186, 49, 120, 9, 100, 11, 12, 1;

%e ...

%e From _Gus Wiseman_, Nov 12 2020: (Start)

%e Row n = 6 counts the following compositions:

%e (6) (15) (114) (1113) (11112) (111111)

%e (51) (123) (1131) (11121)

%e (132) (1311) (11211)

%e (141) (3111) (12111)

%e (213) (21111)

%e (231)

%e (312)

%e (321)

%e (411)

%e (End)

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],Length[#]==1||CoprimeQ@@#&]],{n,10},{k,n}] (* _Gus Wiseman_, Nov 12 2020 *)

%Y A072704 counts the unimodal instead of coprime version.

%Y A087087 and A335235 rank these compositions.

%Y A101268 gives row sums.

%Y A101391 is the relatively prime instead of pairwise coprime version.

%Y A282749 is the unordered version.

%Y A000740 counts relatively prime compositions, with strict case A332004.

%Y A007360 counts pairwise coprime or singleton strict partitions.

%Y A051424 counts pairwise coprime or singleton partitions, ranked by A302569.

%Y A097805 counts compositions by sum and length.

%Y A178472 counts compositions with a common divisor.

%Y A216652 and A072574 count strict compositions by sum and length.

%Y A305713 counts pairwise coprime strict partitions.

%Y A327516 counts pairwise coprime partitions, ranked by A302696.

%Y A335235 ranks pairwise coprime or singleton compositions.

%Y A337462 counts pairwise coprime compositions, ranked by A333227.

%Y A337562 counts pairwise coprime or singleton strict compositions.

%Y A337665 counts compositions whose distinct parts are pairwise coprime, ranked by A333228.

%Y Cf. A000837, A007359, A302568, A337461, A337561, A337667.

%K nonn,tabl

%O 1,5

%A _N. J. A. Sloane_, Mar 05 2017