OFFSET
1,19
COMMENTS
This poset is equivalent to the poset of multiset partitions of the prime indices of n, ordered by refinement.
FORMULA
T(2^n,k) = A330785(n,k).
T(n,1) + T(n,2) = 1.
EXAMPLE
Triangle begins:
1: 16: 0 1 3 2 31: 1 46: 0 1
2: 1 17: 1 32: 0 1 5 8 4 47: 1
3: 1 18: 0 1 2 33: 0 1 48: 0 1 10 23 15
4: 0 1 19: 1 34: 0 1 49: 0 1
5: 1 20: 0 1 2 35: 0 1 50: 0 1 2
6: 0 1 21: 0 1 36: 0 1 7 7 51: 0 1
7: 1 22: 0 1 37: 1 52: 0 1 2
8: 0 1 1 23: 1 38: 0 1 53: 1
9: 0 1 24: 0 1 5 5 39: 0 1 54: 0 1 5 5
10: 0 1 25: 0 1 40: 0 1 5 5 55: 0 1
11: 1 26: 0 1 41: 1 56: 0 1 5 5
12: 0 1 2 27: 0 1 1 42: 0 1 3 57: 0 1
13: 1 28: 0 1 2 43: 1 58: 0 1
14: 0 1 29: 1 44: 0 1 2 59: 1
15: 0 1 30: 0 1 3 45: 0 1 2 60: 0 1 9 11
Row n = 48 counts the following chains (minimum and maximum not shown):
() (6*8) (2*3*8)->(6*8) (2*2*2*6)->(2*4*6)->(6*8)
(2*24) (2*4*6)->(6*8) (2*2*3*4)->(2*3*8)->(6*8)
(3*16) (2*3*8)->(2*24) (2*2*3*4)->(2*4*6)->(6*8)
(4*12) (2*3*8)->(3*16) (2*2*2*6)->(2*4*6)->(2*24)
(2*3*8) (2*4*6)->(2*24) (2*2*2*6)->(2*4*6)->(4*12)
(2*4*6) (2*4*6)->(4*12) (2*2*3*4)->(2*3*8)->(2*24)
(3*4*4) (3*4*4)->(3*16) (2*2*3*4)->(2*3*8)->(3*16)
(2*2*12) (3*4*4)->(4*12) (2*2*3*4)->(2*4*6)->(2*24)
(2*2*2*6) (2*2*12)->(2*24) (2*2*3*4)->(2*4*6)->(4*12)
(2*2*3*4) (2*2*12)->(4*12) (2*2*3*4)->(3*4*4)->(3*16)
(2*2*2*6)->(6*8) (2*2*3*4)->(3*4*4)->(4*12)
(2*2*3*4)->(6*8) (2*2*2*6)->(2*2*12)->(2*24)
(2*2*2*6)->(2*24) (2*2*2*6)->(2*2*12)->(4*12)
(2*2*2*6)->(4*12) (2*2*3*4)->(2*2*12)->(2*24)
(2*2*3*4)->(2*24) (2*2*3*4)->(2*2*12)->(4*12)
(2*2*3*4)->(3*16)
(2*2*3*4)->(4*12)
(2*2*2*6)->(2*4*6)
(2*2*3*4)->(2*3*8)
(2*2*3*4)->(2*4*6)
(2*2*3*4)->(3*4*4)
(2*2*2*6)->(2*2*12)
(2*2*3*4)->(2*2*12)
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
upfacs[q_]:=Union[Sort/@Join@@@Tuples[facs/@q]];
paths[eds_, start_, end_]:=If[start==end, Prepend[#, {}], #]&[Join@@Table[Prepend[#, e]&/@paths[eds, Last[e], end], {e, Select[eds, First[#]==start&]}]];
Table[Length[Select[paths[Join@@Table[{y, #}&/@DeleteCases[upfacs[y], y], {y, facs[n]}], {n}, First[facs[n]]], Length[#]==k-1&]], {n, 100}, {k, PrimeOmega[n]}]
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Jan 04 2020
STATUS
approved