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A282748 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).

%I A282748 #31 Jan 05 2025 19:51:41
%S A282748 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 A282748 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 A282748 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 A282748 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 A282748 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 A282748 Alois P. Heinz, <a href="/A282748/b282748.txt">Rows n = 1..200, flattened</a> (first 100 rows from Chai Wah Wu)
%H A282748 Temba Shonhiwa, <a href="https://web.archive.org/web/2024*/https://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 A282748 It seems that no general formula or recurrence is known, although Shonhiwa gives formulas for a few of the early diagonals.
%e A282748 Triangle begins:
%e A282748   1;
%e A282748   1,  1;
%e A282748   1,  2,  1;
%e A282748   1,  2,  3,   1;
%e A282748   1,  4,  3,   4,   1;
%e A282748   1,  2,  9,   4,   5,   1;
%e A282748   1,  6,  3,  16,   5,   6,  1;
%e A282748   1,  4, 15,   4,  25,   6,  7,   1;
%e A282748   1,  6,  9,  28,   5,  36,  7,   8,  1;
%e A282748   1,  4, 21,  16,  45,   6, 49,   8,  9,   1;
%e A282748   1, 10,  9,  52,  25,  66,  7,  64,  9,  10,  1;
%e A282748   1,  4, 39,  16, 105,  36, 91,   8, 81,  10, 11,  1;
%e A282748   1, 12,  9, 100,  25, 186, 49, 120,  9, 100, 11, 12, 1;
%e A282748   ...
%e A282748 From _Gus Wiseman_, Nov 12 2020: (Start)
%e A282748 Row n = 6 counts the following compositions:
%e A282748   (6)  (15)  (114)  (1113)  (11112)  (111111)
%e A282748        (51)  (123)  (1131)  (11121)
%e A282748              (132)  (1311)  (11211)
%e A282748              (141)  (3111)  (12111)
%e A282748              (213)          (21111)
%e A282748              (231)
%e A282748              (312)
%e A282748              (321)
%e A282748              (411)
%e A282748 (End)
%t A282748 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],Length[#]==1||CoprimeQ@@#&]],{n,10},{k,n}] (* _Gus Wiseman_, Nov 12 2020 *)
%Y A282748 A072704 counts the unimodal instead of coprime version.
%Y A282748 A087087 and A335235 rank these compositions.
%Y A282748 A101268 gives row sums.
%Y A282748 A101391 is the relatively prime instead of pairwise coprime version.
%Y A282748 A282749 is the unordered version.
%Y A282748 A000740 counts relatively prime compositions, with strict case A332004.
%Y A282748 A007360 counts pairwise coprime or singleton strict partitions.
%Y A282748 A051424 counts pairwise coprime or singleton partitions, ranked by A302569.
%Y A282748 A097805 counts compositions by sum and length.
%Y A282748 A178472 counts compositions with a common divisor.
%Y A282748 A216652 and A072574 count strict compositions by sum and length.
%Y A282748 A305713 counts pairwise coprime strict partitions.
%Y A282748 A327516 counts pairwise coprime partitions, ranked by A302696.
%Y A282748 A335235 ranks pairwise coprime or singleton compositions.
%Y A282748 A337462 counts pairwise coprime compositions, ranked by A333227.
%Y A282748 A337562 counts pairwise coprime or singleton strict compositions.
%Y A282748 A337665 counts compositions whose distinct parts are pairwise coprime, ranked by A333228.
%Y A282748 Cf. A000837, A007359, A302568, A337461, A337561, A337667.
%K A282748 nonn,tabl
%O A282748 1,5
%A A282748 _N. J. A. Sloane_, Mar 05 2017