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A374252 Irregular triangle read by rows where T(n,k) is the number of permutations of the prime factors of n with k runs.

%I A374252 #5 Jul 08 2024 16:40:51
%S A374252 1,1,1,0,1,0,2,1,1,0,0,1,0,0,2,1,0,2,1,1,0,2,0,2,1,0,0,0,1,0,2,1,1,0,
%T A374252 2,1,0,2,0,2,1,0,2,2,0,1,0,0,2,1,0,0,0,2,1,1,0,0,6,1,1,0,0,0,0,0,2,0,
%U A374252 2,0,2,0,2,2,2,1,0,2,0,2,0,2,2,0,1
%N A374252 Irregular triangle read by rows where T(n,k) is the number of permutations of the prime factors of n with k runs.
%C A374252 An alternative form of this sequence (with the same data) has offset 1 and begins with an empty row.
%C A374252 Note that the prime factors of n are separable (A335433) iff the last term of row n is positive.
%e A374252 The T(36,3) = 2 permutations are (2,3,3,2) and (3,2,2,3).
%e A374252 Row n = 72 counts the following permutations:
%e A374252   .  (2,2,2,3,3)  (2,2,3,3,2)  (2,2,3,2,3)  (2,3,2,3,2)
%e A374252      (3,3,2,2,2)  (2,3,3,2,2)  (2,3,2,2,3)
%e A374252                   (3,2,2,2,3)  (3,2,2,3,2)
%e A374252                                (3,2,3,2,2)
%e A374252 Triangle begins:
%e A374252    1:
%e A374252    2: 1
%e A374252    3: 1
%e A374252    4: 1  0
%e A374252    5: 1
%e A374252    6: 0  2
%e A374252    7: 1
%e A374252    8: 1  0  0
%e A374252    9: 1  0
%e A374252   10: 0  2
%e A374252   11: 1
%e A374252   12: 0  2  1
%e A374252   13: 1
%e A374252   14: 0  2
%e A374252   15: 0  2
%e A374252   16: 1  0  0  0
%e A374252   17: 1
%e A374252   18: 0  2  1
%e A374252   19: 1
%e A374252   20: 0  2  1
%t A374252 prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
%t A374252 Table[Length[Select[Permutations[prifacs[n]], Length[Split[#]]==k&]],{n,100},{k,PrimeOmega[n]}]
%Y A374252 Row-lengths are A001222.
%Y A374252 Row-sums are A008480 (number of permutations of prime factors).
%Y A374252 Column k = 1 is A069513.
%Y A374252 For compositions instead of permutations of prime factors we have A238130.
%Y A374252 Last column is A335452 (where k = A001222(n)), which counts separations.
%Y A374252 Position of the last positive term in row n is A373957(n).
%Y A374252 The number of zeros at the end of row n is A374246(n).
%Y A374252 The number of nonzero terms in row n is A374247(n).
%Y A374252 A001221 counts distinct prime factors.
%Y A374252 A003242 counts run-compressed compositions, i.e., anti-runs.
%Y A374252 A124767 counts runs in standard compositions, anti-runs A333381.
%Y A374252 A333755 counts compositions by number of runs.
%Y A374252 A335433 lists separable numbers, complement A335448.
%Y A374252 A374250 maximizes sum of run-compression, for indices A373956.
%Y A374252 Cf. A026549, A027748, A046660, A126706, A151821, A246655.
%K A374252 nonn,tabf
%O A374252 2,7
%A A374252 _Gus Wiseman_, Jul 07 2024