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Irregular triangle read by rows: T(n, m) is the number of ways to distribute Omega(n) objects into precisely m distinct boxes, with no box empty (Omega(n) >= m).
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%I #40 Oct 29 2020 09:48:23

%S 0,0,1,0,1,0,1,1,0,1,0,1,2,0,1,0,1,2,1,0,1,1,0,1,2,0,1,0,1,4,3,0,1,0,

%T 1,2,0,1,2,0,1,3,3,1,0,1,0,1,4,3,0,1,0,1,4,3,0,1,2,0,1,2,0,1,0,1,6,9,

%U 4,0,1,1,0,1,2,0,1,2,1,0,1,4,3,0,1,0,1,6,6

%N Irregular triangle read by rows: T(n, m) is the number of ways to distribute Omega(n) objects into precisely m distinct boxes, with no box empty (Omega(n) >= m).

%C n is the specification number for a set of Omega(n) objects (see Theorem 3 in Beekman's article).

%C The specification number of a multiset is also called its Heinz number. - _Gus Wiseman_, Aug 25 2020

%C From _Gus Wiseman_, Aug 25 2020: (Start)

%C For n > 1, T(n,k) is also the number of ordered factorizations of n into k factors > 1. For example, row n = 24 counts the following ordered factorizations (the first column is empty):

%C 24 3*8 2*2*6 2*2*2*3

%C 4*6 2*3*4 2*2*3*2

%C 6*4 2*4*3 2*3*2*2

%C 8*3 2*6*2 3*2*2*2

%C 12*2 3*2*4

%C 2*12 3*4*2

%C 4*2*3

%C 4*3*2

%C 6*2*2

%C For n > 1, T(n,k) is also the number of strict length-k chains of divisors from n to 1. For example, row n = 36 counts the following chains (the first column is empty):

%C 36/1 36/2/1 36/4/2/1 36/12/4/2/1

%C 36/3/1 36/6/2/1 36/12/6/2/1

%C 36/4/1 36/6/3/1 36/12/6/3/1

%C 36/6/1 36/9/3/1 36/18/6/2/1

%C 36/9/1 36/12/2/1 36/18/6/3/1

%C 36/12/1 36/12/3/1 36/18/9/3/1

%C 36/18/1 36/12/4/1

%C 36/12/6/1

%C 36/18/2/1

%C 36/18/3/1

%C 36/18/6/1

%C 36/18/9/1

%C (End)

%D Richard Beekman, An Introduction to Number-Theoretic Combinatorics, Lulu Press 2017.

%H Stefano Spezia, <a href="/A334996/b334996.txt">First 3000 rows of the table, flattened</a>

%H Richard Beekman, <a href="https://www.researchgate.net/publication/341090354_A_General_Solution_of_the_Ferris_Wheel_Problem">A General Solution of the Ferris Wheel Problem</a>.

%F T(n, m) = Sum_{k=0..m-1} (-1)^k*binomial(m,k)*tau_{m-k-1}(n), where tau_s(r) = A334997(r, s) (see Theorem 3, Lemma 1 and Lemma 2 in Beekman's article).

%F Conjecture: Sum_{m=0..Omega(n)} T(n, m) = A002033(n-1) for n > 1.

%e The triangle T(n, m) begins

%e n\m| 0 1 2 3 4

%e ---+--------------------------

%e 1 | 0

%e 2 | 0 1

%e 3 | 0 1

%e 4 | 0 1 1

%e 5 | 0 1

%e 6 | 0 1 2

%e 7 | 0 1

%e 8 | 0 1 2 1

%e 9 | 0 1 1

%e 10 | 0 1 2

%e 11 | 0 1

%e 12 | 0 1 4 3

%e 13 | 0 1

%e 14 | 0 1 2

%e 15 | 0 1 2

%e 16 | 0 1 3 3 1

%e ...

%e From _Gus Wiseman_, Aug 25 2020: (Start)

%e Row n = 36 counts the following distributions of {1,1,2,2} (the first column is empty):

%e {1122} {1}{122} {1}{1}{22} {1}{1}{2}{2}

%e {11}{22} {1}{12}{2} {1}{2}{1}{2}

%e {112}{2} {11}{2}{2} {1}{2}{2}{1}

%e {12}{12} {1}{2}{12} {2}{1}{1}{2}

%e {122}{1} {12}{1}{2} {2}{1}{2}{1}

%e {2}{112} {1}{22}{1} {2}{2}{1}{1}

%e {22}{11} {12}{2}{1}

%e {2}{1}{12}

%e {2}{11}{2}

%e {2}{12}{1}

%e {2}{2}{11}

%e {22}{1}{1}

%e (End)

%t tau[n_,k_]:=If[n==1,1,Product[Binomial[Extract[Extract[FactorInteger[n],i],2]+k,k],{i,1,Length[FactorInteger[n]]}]]; (* A334997 *)

%t T[n_,m_]:=Sum[(-1)^k*Binomial[m,k]*tau[n,m-k-1],{k,0,m-1}]; Table[T[n,m],{n,1,30},{m,0,PrimeOmega[n]}]//Flatten

%t (* second program *)

%t chc[n_]:=If[n==1,{{}},Prepend[Join@@Table[Prepend[#,n]&/@chc[d],{d,DeleteCases[Divisors[n],1|n]}],{n}]]; (* change {{}} to {} if a(1) = 0 *)

%t Table[Length[Select[chc[n],Length[#]==k&]],{n,30},{k,0,PrimeOmega[n]}] (* _Gus Wiseman_, Aug 25 2020 *)

%o (PARI) TT(n, k) = if (k==0, 1, sumdiv(n, d, TT(d, k-1))); \\ A334997

%o T(n, m) = sum(k=0, m-1, (-1)^k*binomial(m, k)*TT(n, m-k-1));

%o tabf(nn) = {for (n=1, nn, print(vector(bigomega(n)+1, k, T(n, k-1))););} \\ _Michel Marcus_, May 20 2020

%Y Cf. A000007 (1st column), A000012 (2nd column), A001222 (Omega function), A002033 (row sums shifted left), A007318.

%Y A008480 gives rows ends.

%Y A073093 gives row lengths.

%Y A074206 gives row sums.

%Y A112798 constructs the multiset with each specification number.

%Y A124433 is a signed version.

%Y A251683 is the version with zeros removed.

%Y A334997 is the non-strict version.

%Y A337107 is the restriction to factorial numbers.

%Y A001055 counts factorizations.

%Y A067824 counts strict chains of divisors starting with n.

%Y A122651 counts strict chains of divisors summing to n.

%Y A167865 counts strict chains of divisors > 1 summing to n.

%Y A253249 counts strict chains of divisors.

%Y A337105 counts strict chains of divisors from n! to 1.

%Y Cf. A007425, A008683, A056239, A124010, A167865, A317144, A319001.

%K nonn,tabf

%O 1,13

%A _Stefano Spezia_, May 19 2020