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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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,19 COMMENTS This poset is equivalent to the poset of multiset partitions of the prime indices of n, ordered by refinement. LINKS 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. Cf. A001055, A002846, A003238, A007716, A281118, A292504, A292505, A318812, A330665, A330727. Sequence in context: A118777 A073068 A166006 * A208769 A255327 A255391 Adjacent sequences:  A330932 A330933 A330934 * A330936 A330937 A330938 KEYWORD nonn,tabf AUTHOR Gus Wiseman, Jan 04 2020 STATUS approved

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Last modified July 27 13:22 EDT 2021. Contains 346306 sequences. (Running on oeis4.)