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A323719
Array read by antidiagonals upwards where A(n, k) is the number of orderless factorizations of n with k - 1 levels of parentheses.
3
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 3, 1, 4, 1, 6, 1, 1, 1, 1, 2, 6, 1, 5, 1, 7, 1, 1, 1, 1, 2, 3, 10, 1, 6, 1, 8, 1, 1, 1, 1, 1, 3, 4, 15, 1, 7, 1, 9, 1, 1, 1, 1, 4, 1, 4, 5, 21, 1, 8, 1, 10, 1, 1, 1
OFFSET
1,12
COMMENTS
An orderless factorization of n with k > 1 levels of parentheses is any multiset partition of an orderless factorization of n with k - 1 levels of parentheses. If k = 1 it is just an orderless factorization of n into factors > 1.
EXAMPLE
Array begins:
k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k=10 k=11 k=12
n=1: 1 1 1 1 1 1 1 1 1 1 1 1 1
n=2: 1 1 1 1 1 1 1 1 1 1 1 1 1
n=3: 1 1 1 1 1 1 1 1 1 1 1 1 1
n=4: 1 2 3 4 5 6 7 8 9 10 11 12 13
n=5: 1 1 1 1 1 1 1 1 1 1 1 1 1
n=6: 1 2 3 4 5 6 7 8 9 10 11 12 13
n=7: 1 1 1 1 1 1 1 1 1 1 1 1 1
n=8: 1 3 6 10 15 21 28 36 45 55 66 78 91
n=9: 1 2 3 4 5 6 7 8 9 10 11 12 13
n=10: 1 2 3 4 5 6 7 8 9 10 11 12 13
n=11: 1 1 1 1 1 1 1 1 1 1 1 1 1
n=12: 1 4 9 16 25 36 49 64 81 100 121 144 169
n=13: 1 1 1 1 1 1 1 1 1 1 1 1 1
n=14: 1 2 3 4 5 6 7 8 9 10 11 12 13
n=15: 1 2 3 4 5 6 7 8 9 10 11 12 13
n=16: 1 5 14 30 55 91 140 204 285 385 506 650 819
n=17: 1 1 1 1 1 1 1 1 1 1 1 1 1
n=18: 1 4 9 16 25 36 49 64 81 100 121 144 169
The A(12,3) = 16 orderless factorizations of 12 with 2 levels of parentheses:
((2*2*3)) ((2*6)) ((3*4)) ((12))
((2)*(2*3)) ((2)*(6)) ((3)*(4))
((3)*(2*2)) ((2))*((6)) ((3))*((4))
((2))*((2*3))
((2)*(2)*(3))
((3))*((2*2))
((2))*((2)*(3))
((3))*((2)*(2))
((2))*((2))*((3))
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
lev[n_, k_]:=If[k==0, {n}, Join@@Table[Union[Sort/@Tuples[lev[#, k-1]&/@fac]], {fac, facs[n]}]];
Table[Length[lev[sum-k, k]], {sum, 12}, {k, 0, sum-1}]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jan 25 2019
STATUS
approved