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A330935
Irregular triangle read by rows where T(n,k) is the number of length-k chains from minimum to maximum in the poset of factorizations of n into factors > 1, ordered by refinement.
10
1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 2, 1, 0, 1, 2, 0, 1, 0, 1, 1, 0, 1, 5, 5, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 1, 0, 1, 5, 8, 4, 0, 1, 0, 1, 0, 1, 0, 1, 7, 7, 1, 0, 1, 0, 1, 0, 1, 5, 5, 1, 0, 1
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
Row lengths are A001222.
Row sums are A317176.
Column k = 1 is A010051.
Column k = 2 is A066247.
Column k = 3 is A330936.
Final terms of each row are A317145.
The version for set partitions is A008826, with row sums A005121.
The version for integer partitions is A330785, with row sums A213427.
Sequence in context: A118777 A073068 A166006 * A208769 A255327 A255391
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Jan 04 2020
STATUS
approved