# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a282748 Showing 1-1 of 1 %I A282748 #29 Nov 12 2020 22:18:36 %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, Rows n = 1..200, flattened (first 100 rows from Chai Wah Wu) %H A282748 Temba Shonhiwa, Compositions with pairwise relatively prime summands within a restricted setting, 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 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE