A107106 Divide A036039(n) by A036040(n).

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, 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, 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

Offset: 1

Author

Alford Arnold, May 12 2005

Keywords

• easy
• nonn
• tabf

Comments

A107107 gives the row sums. - R. J. Mathar, Aug 13 2007

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.

Examples

			a(36) = 280/70 = 4.
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];...
		

Links

• Jean-François Alcover after Wolfdieter Lang, Table of n, a(n) for n = 1..138
• Wolfdieter Lang, First 10 rows of the array and more.

Crossrefs

Cf. A107107.

Programs

Maple

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

Mathematica

aspartitions[n_] := Reverse /@ Sort[Sort /@ IntegerPartitions[n]];
A036039[n_] := n!/(Times @@ #)& /@ ((#! Range[n]^#)& /@ Function[par, Count[par, #]& /@ Range[n]] /@ aspartitions[n]);
runs[li : {__Integer}] := ((Length /@ Split[#]))&[Sort@li];
A036040[n_] := Module[{temp}, temp = Map[Reverse, Sort@(Sort /@ IntegerPartitions[n]), {1}]; Apply[Multinomial, temp, {1}]/Apply[Times, (runs /@ temp)!, {1}]];
T[n_] := A036039[n]/A036040[n];
Table[T[n], {n, 1, 10}] // Flatten
(* Jean-François Alcover, Jun 10 2023, after Wouter Meeussen in A036039 *)

Formula

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

Extensions

Corrected and extended by R. J. Mathar, Aug 13 2007

a(75) and a(76) swapped (first 36, then 24) by Wolfdieter Lang, Sep 22 2008