A107106 Irregular triangle T(n,k) = A036039(n,k) / A036040(n,k), read by rows, 1 <= k <= A000041(n).

1, 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

0,5

Comments

Rows have A000041(n) entries, with partitions in Abramowitz and Stegun order (A036036).

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20; rows 1..10 from Jean-François Alcover)

Wolfdieter Lang, First 10 rows of the array and more.

Example

Triangle begins:
  0 | 1;
  1 | 1;
  2 | 1, 1;
  3 | 2, 1, 1;
  4 | 6, 2, 1, 1, 1;
  5 | 24, 6, 2, 2, 1, 1, 1;
  6 | 120, 24, 6, 4, 6, 2, 1, 2, 1, 1, 1;
  7 | 720, 120, 24, 12, 24, 6, 4, 2, 6, 2, 1, 2, 1, 1, 1;
  ...

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 *)

Prog

(PARI)
C(sig)={prod(k=1, #sig, sig[k]!)/vecprod(sig)}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 03 2025

Crossrefs

Cf. A000041 (row lengths), A036036, A036039, A036040, A107107 (row sums), A134133.

Sequence in context: A123361 A265315 A179380 * A178249 A114423 A119502

Adjacent sequences: A107103 A107104 A107105 * A107107 A107108 A107109

Keyword

easy,nonn,tabf

Author

Alford Arnold, May 12 2005

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

a(0)=1 prepended by Andrew Howroyd, Oct 03 2025

Status

approved