|
%I #6 Jan 05 2020 08:11:12
%S 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,
%T 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,
%U 1,0,1,0,1,7,7,1,0,1,0,1,0,1,5,5,1,0,1
%N 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.
%C This poset is equivalent to the poset of multiset partitions of the prime indices of n, ordered by refinement.
%F T(2^n,k) = A330785(n,k).
%F T(n,1) + T(n,2) = 1.
%e Triangle begins:
%e 1: 16: 0 1 3 2 31: 1 46: 0 1
%e 2: 1 17: 1 32: 0 1 5 8 4 47: 1
%e 3: 1 18: 0 1 2 33: 0 1 48: 0 1 10 23 15
%e 4: 0 1 19: 1 34: 0 1 49: 0 1
%e 5: 1 20: 0 1 2 35: 0 1 50: 0 1 2
%e 6: 0 1 21: 0 1 36: 0 1 7 7 51: 0 1
%e 7: 1 22: 0 1 37: 1 52: 0 1 2
%e 8: 0 1 1 23: 1 38: 0 1 53: 1
%e 9: 0 1 24: 0 1 5 5 39: 0 1 54: 0 1 5 5
%e 10: 0 1 25: 0 1 40: 0 1 5 5 55: 0 1
%e 11: 1 26: 0 1 41: 1 56: 0 1 5 5
%e 12: 0 1 2 27: 0 1 1 42: 0 1 3 57: 0 1
%e 13: 1 28: 0 1 2 43: 1 58: 0 1
%e 14: 0 1 29: 1 44: 0 1 2 59: 1
%e 15: 0 1 30: 0 1 3 45: 0 1 2 60: 0 1 9 11
%e Row n = 48 counts the following chains (minimum and maximum not shown):
%e () (6*8) (2*3*8)->(6*8) (2*2*2*6)->(2*4*6)->(6*8)
%e (2*24) (2*4*6)->(6*8) (2*2*3*4)->(2*3*8)->(6*8)
%e (3*16) (2*3*8)->(2*24) (2*2*3*4)->(2*4*6)->(6*8)
%e (4*12) (2*3*8)->(3*16) (2*2*2*6)->(2*4*6)->(2*24)
%e (2*3*8) (2*4*6)->(2*24) (2*2*2*6)->(2*4*6)->(4*12)
%e (2*4*6) (2*4*6)->(4*12) (2*2*3*4)->(2*3*8)->(2*24)
%e (3*4*4) (3*4*4)->(3*16) (2*2*3*4)->(2*3*8)->(3*16)
%e (2*2*12) (3*4*4)->(4*12) (2*2*3*4)->(2*4*6)->(2*24)
%e (2*2*2*6) (2*2*12)->(2*24) (2*2*3*4)->(2*4*6)->(4*12)
%e (2*2*3*4) (2*2*12)->(4*12) (2*2*3*4)->(3*4*4)->(3*16)
%e (2*2*2*6)->(6*8) (2*2*3*4)->(3*4*4)->(4*12)
%e (2*2*3*4)->(6*8) (2*2*2*6)->(2*2*12)->(2*24)
%e (2*2*2*6)->(2*24) (2*2*2*6)->(2*2*12)->(4*12)
%e (2*2*2*6)->(4*12) (2*2*3*4)->(2*2*12)->(2*24)
%e (2*2*3*4)->(2*24) (2*2*3*4)->(2*2*12)->(4*12)
%e (2*2*3*4)->(3*16)
%e (2*2*3*4)->(4*12)
%e (2*2*2*6)->(2*4*6)
%e (2*2*3*4)->(2*3*8)
%e (2*2*3*4)->(2*4*6)
%e (2*2*3*4)->(3*4*4)
%e (2*2*2*6)->(2*2*12)
%e (2*2*3*4)->(2*2*12)
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t upfacs[q_]:=Union[Sort/@Join@@@Tuples[facs/@q]];
%t paths[eds_,start_,end_]:=If[start==end,Prepend[#,{}],#]&[Join@@Table[Prepend[#,e]&/@paths[eds,Last[e],end],{e,Select[eds,First[#]==start&]}]];
%t 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]}]
%Y Row lengths are A001222.
%Y Row sums are A317176.
%Y Column k = 1 is A010051.
%Y Column k = 2 is A066247.
%Y Column k = 3 is A330936.
%Y Final terms of each row are A317145.
%Y The version for set partitions is A008826, with row sums A005121.
%Y The version for integer partitions is A330785, with row sums A213427.
%Y Cf. A001055, A002846, A003238, A007716, A281118, A292504, A292505, A318812, A330665, A330727.
%K nonn,tabf
%O 1,19
%A _Gus Wiseman_, Jan 04 2020
|
