login
Divide A036039(n) by A036040(n).
4

%I #23 Jun 11 2023 12:29:52

%S 1,1,1,2,1,1,6,2,1,1,1,24,6,2,2,1,1,1,120,24,6,4,6,2,1,2,1,1,1,720,

%T 120,24,12,24,6,4,2,6,2,1,2,1,1,1,5040,720,120,48,36,120,24,12,6,4,24,

%U 6,4,2,1,6,2,1,2,1,1,1,40320,5040,720,240,144,720,120,48,36,24,12,8,120,24,12

%N Divide A036039(n) by A036040(n).

%C A107107 gives the row sums. - _R. J. Mathar_, Aug 13 2007

%C This array is the first one (K=1) of a family of partition number arrays called M31(1). For M31(2) see A134133 = M_3(2)/M_3.

%H Jean-François Alcover after Wolfdieter Lang, <a href="/A107106/b107106.txt">Table of n, a(n) for n = 1..138</a>

%H Wolfdieter Lang, <a href="/A107106/a107106.txt">First 10 rows of the array and more</a>.

%F a(n) = A036039(n) / A036040(n).

%e a(36) = 280/70 = 4.

%e As array: [1];[1,1];[2,1,1];[6,2,1,1,1];[24,6,2,2,1,1,1];[120,24,6,4,6,2,1,2,1,1,1];...

%p sortAbrSteg := proc(L1,L2) local i ; if nops(L1) < nops(L2) then RETURN(true) ; elif nops(L2) < nops(L1) then RETURN(false) ; else for i from 1 to nops(L1) do if op(i,L1) < op(i,L2) then RETURN(false) ; fi ; od ; RETURN(true) ; fi ; end: M2overM3 := proc(L) local n,k,an,resul; n := add(i,i=L) ; resul := 1 ; for k from 1 to n do an := add(1-min(abs(j-k),1),j=L) ; resul := resul* (factorial(k-1))^an ; od ; end: A107106 := proc(n,k) local prts,m ; prts := combinat[partition](n) ; prts := sort(prts, sortAbrSteg) ; if k <= nops(prts) then M2overM3(op(k,prts)) ; else 0 ; fi ; end: for n from 1 to 10 do for k from 1 to combinat[numbpart](n) do a:=A107106(n,k) ; printf("%d,",a) ; od; od ; # _R. J. Mathar_, Aug 13 2007

%t aspartitions[n_] := Reverse /@ Sort[Sort /@ IntegerPartitions[n]];

%t A036039[n_] := n!/(Times @@ #)& /@ ((#! Range[n]^#)& /@ Function[par, Count[par, #]& /@ Range[n]] /@ aspartitions[n]);

%t runs[li : {__Integer}] := ((Length /@ Split[#]))&[Sort@li];

%t A036040[n_] := Module[{temp}, temp = Map[Reverse, Sort@(Sort /@ IntegerPartitions[n]), {1}]; Apply[Multinomial, temp, {1}]/Apply[Times, (runs /@ temp)!, {1}]];

%t T[n_] := A036039[n]/A036040[n];

%t Table[T[n], {n, 1, 10}] // Flatten

%t (* _Jean-François Alcover_, Jun 10 2023, after _Wouter Meeussen_ in A036039 *)

%Y Cf. A107107.

%K easy,nonn,tabf

%O 1,4

%A _Alford Arnold_, May 12 2005

%E Corrected and extended by _R. J. Mathar_, Aug 13 2007

%E a(75) and a(76) swapped (first 36, then 24) by _Wolfdieter Lang_, Sep 22 2008