A080577 -id:A080577 - cp's OEIS frontend

A063008 Canonical partition sequence (see A080577) encoded by prime factorization. The partition [p1,p2,p3,...] with p1 >= p2 >= p3 >= ... is encoded as 2^p1 * 3^p2 * 5^p3 * ... .

1, 2, 4, 6, 8, 12, 30, 16, 24, 36, 60, 210, 32, 48, 72, 120, 180, 420, 2310, 64, 96, 144, 240, 216, 360, 840, 900, 1260, 4620, 30030, 128, 192, 288, 480, 432, 720, 1680, 1080, 1800, 2520, 9240, 6300, 13860, 60060, 510510, 256, 384, 576, 960, 864, 1440, 3360

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 02 2001

Partitions are ordered first by sum. Then all partitions of n are viewed as exponent tuples on n variables and their corresponding monomials are ordered reverse lexicographically. This gives a canonical ordering: [] [1] [2,0] [1,1] [3,0,0] [2,1,0] [1,1,1] [4,0,0,0] [3,1,0,0] [2,2,0,0] [2,1,1,0] [1,1,1,1]... Rearrangement of A025487, A036035 etc.

Or, least integer of each prime signature; resorted in accordance with the integer partitions described in A080577. - Alford Arnold, Feb 13 2008

			Partition [2,1,1,1] for n=5 gives 2^2*3*5*7 = 420.
The sequence begins:
   1;
   2;
   4,  6;
   8, 12,  30;
  16, 24,  36,  60, 210;
  32, 48,  72, 120, 180, 420, 2310;
  64, 96, 144, 240, 216, 360,  840, 900, 1260, 4620, 30030;
  ...

Alois P. Heinz, Rows n = 0..30, flattened
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arXiv:1405.5283 [math.NT], 2014.

Cf. A001222 (bigomega), A025487, A059901.
See A080576 Maple (graded reflected lexicographic) ordering.
See A080577 Mathematica (graded reverse lexicographic) ordering.
See A036036 "Abramowitz and Stegun" (graded reflected colexicographic) ordering.
See A036037 for graded colexicographic ordering.

with(combinat): A063008_row := proc(n) local e,w,r;
r := proc(L) local B,i; B := NULL;
for i from nops(L) by -1 to 1 do
B := B,L[i] od; [%] end:
w := proc(e) local i, m, p, P; m := infinity;
P := permute([seq(ithprime(i),i=1..nops(e))]);
for p in P do m := min(m,mul(p[i]^e[i],i=1..nops(e))) od end:
[seq(w(e), e = r(partition(n)))] end:
seq(print(A063008_row(i)),i=0..6); # Peter Luschny, Jan 23 2011
# second Maple program:
b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
    [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> map(x-> mul(ithprime(i)^x[i], i=1..nops(x)), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Sep 03 2019

row[n_] := Product[ Prime[k]^#[[k]], {k, 1, Length[#]}]& /@ IntegerPartitions[n]; Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Dec 10 2012 *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {Table[1, {n}]},Join[ Prepend[#, i]& /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
T[n_] := Product[Prime[i]^#[[i]], {i, 1, Length[#]}]& /@ b[n, n];
T /@ Range[0, 9] // Flatten (* Jean-François Alcover, Jun 09 2021, after Alois P. Heinz *)

bigomega(T(n,k)) = n. - Andrew Howroyd, Mar 28 2020

Partially edited by N. J. A. Sloane, May 15, at the suggestion of R. J. Mathar

Corrected and (minor) edited by Daniel Forgues, Jan 03 2011

A331581 Maximum part of the n-th integer partition in graded reverse-lexicographic order (A080577); a(1) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 1, 4, 3, 2, 2, 1, 5, 4, 3, 3, 2, 2, 1, 6, 5, 4, 4, 3, 3, 3, 2, 2, 2, 1, 7, 6, 5, 5, 4, 4, 4, 3, 3, 3, 3, 2, 2, 2, 1, 8, 7, 6, 6, 5, 5, 5, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 1, 9, 8, 7, 7, 6, 6, 6, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 1

Offset: 1

Gus Wiseman, May 08 2020

The first partition ranked by A080577 is (); there is no zeroth partition.

			The sequence of all partitions in graded reverse-lexicographic order begins as follows. The terms are the initial parts.
  ()         (3,2)        (2,1,1,1,1)    (2,2,1,1,1)
  (1)        (3,1,1)      (1,1,1,1,1,1)  (2,1,1,1,1,1)
  (2)        (2,2,1)      (7)            (1,1,1,1,1,1,1)
  (1,1)      (2,1,1,1)    (6,1)          (8)
  (3)        (1,1,1,1,1)  (5,2)          (7,1)
  (2,1)      (6)          (5,1,1)        (6,2)
  (1,1,1)    (5,1)        (4,3)          (6,1,1)
  (4)        (4,2)        (4,2,1)        (5,3)
  (3,1)      (4,1,1)      (4,1,1,1)      (5,2,1)
  (2,2)      (3,3)        (3,3,1)        (5,1,1,1)
  (2,1,1)    (3,2,1)      (3,2,2)        (4,4)
  (1,1,1,1)  (3,1,1,1)    (3,2,1,1)      (4,3,1)
  (5)        (2,2,2)      (3,1,1,1,1)    (4,2,2)
  (4,1)      (2,2,1,1)    (2,2,2,1)      (4,2,1,1)
Triangle begins:
  0
  1
  2 1
  3 2 1
  4 3 2 2 1
  5 4 3 3 2 2 1
  6 5 4 4 3 3 3 2 2 2 1
  7 6 5 5 4 4 4 3 3 3 3 2 2 2 1
  8 7 6 6 5 5 5 4 4 4 4 4 3 3 3 3 3 2 2 2 2 1

OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
Lexicographically ordered reversed partitions are A026791.
Reverse-colexicographically ordered partitions are A026792.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
The version for compositions is A065120 or A333766.
Reverse-lexicographically ordered partitions are A080577.
Distinct parts of these partitions are counted by A115623.
Lexicographically ordered partitions are A193073.
Colexicographically ordered partitions are A211992.
Lengths of these partitions are A238966.
Cf. A036037, A048793, A063008, A066099, A129129, A185974, A228100, A228531, A334301, A334434, A334436, A334438.

revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];
Prepend[First/@Join@@Table[Sort[IntegerPartitions[n],revlexsort],{n,8}],0]

a(n) = A061395(A129129(n - 1)).

A138151 Irregular triangle read by rows in which rows 1..n (when read together) list all the parts in the partitions of n and row n starts with the partitions of n that do not contain 1 as a part (in the order used for A080577).

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 2, 2, 1, 1, 1, 5, 3, 2, 1, 1, 1, 1, 1, 6, 4, 2, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 7, 5, 2, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 6, 2, 5, 3, 4, 4, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 7, 2, 6, 3, 5, 4, 5, 2, 2, 4, 3, 2, 3, 3, 3, 3, 2, 2

Offset: 1

Omar E. Pol, Mar 21 2008

The remainder of row n is necessarily A000041(n-1) 1's.
Previous name: A shell model of partitions. Row n lists the parts of the last section of the set of partitions of n.
Row n lists the nonzero terms of the n-th row of A138136 together with A000041(n-1) 1's.
Row n is also the n-th row of A138138 in reverse order.

			Triangle begins:
1
2,1
3,1,1
4,2,2,1,1,1
5,3,2,1,1,1,1,1,
6,4,2,3,3,2,2,2,1,1,1,1,1,1,1
7,5,2,4,3,3,2,2,1,1,1,1,1,1,1,1,1,1,1
8,6,2,5,3,4,4,4,2,2,3,3,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
9,7,2,6,3,5,4,5,2,2,4,3,2,3,3,3,3,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

Robert Price, Table of n, a(n) for n = 1..4630, 20 rows.
Omar E. Pol, Illustration of the rows 1..6 of the triangle

Mirror of A138138.
Row lengths give A138137.
Row sums give A138879.
Column 1 gives A000027.
Right border gives A000012.
Another version is A138121 which is the mirror of A135010.
Cf. A000041, A080577, A138136, A139094, A139100.

Table[Cases[IntegerPartitions[n], x_ /; Last[x] != 1] ~Join~ConstantArray[{1}, PartitionsP[n - 1]], {n, 8}] // Flatten (* Robert Price, May 22 2020 *)

New name and comments edited by Peter Munn and Omar E. Pol, Jul 25 2025

A131822 Increment each prime factor for each term of the least prime signature sequence derived from A080577.

Original entry on oeis.org

1, 3, 9, 15, 27, 45, 105, 81, 135, 225, 315, 1155, 243, 405, 675, 945, 1575, 3465, 15015, 729, 1215, 2025, 2835, 3375, 4725, 10395, 11025, 17325, 45045, 255255, 2187, 3645, 6075, 8505, 10125, 14175, 31185, 23625, 33075, 51975, 135135, 121275, 225225

Offset: 1

Alford Arnold, Jul 19 2007

			The term 30 = 2*3*5 becomes 105 = 3*5*7.
From A080577 we obtain
   1
   2
   4,  6
   8, 12, 30
  16, 24, 36, 60, ...
  etc.
so the sequence begins
   1
   3
   9,  15
  27,  45, 105
  81, 135, 225, 315, ...
  etc.

Cf. A080577, A131801.

A003961 := proc(n) local ifs,i ; ifs := ifactors(n)[2] ; mul(nextprime(op(1,i))^op(2,i), i=ifs) ; end: A036042 := proc(n) local a, nredu ; a := 0 ; nredu := n+1 ; while nredu > 0 do nredu := nredu-combinat[numbpart](a) ; a := a+1 ; od: RETURN(a-1) ; end: A036035 := proc(n) local row,idx,pa,a,i ; if n = 0 then 1 ; else row := A036042(n) ; idx := n-add(combinat[numbpart](i),i=0..row-1) ; pa := op(-idx-1,combinat[partition](row)) ; a := 1; for i from 1 to nops(pa) do a := a*ithprime(i)^op(-i,pa) ; od; RETURN(a) ; fi ; end: A131822 := proc(n) A003961(A036035(n-1)) ; end: seq(A131822(n),n=1..80) ; # R. J. Mathar, Nov 11 2007

a(n) = A003961(A036035(n-1)). - R. J. Mathar, Nov 11 2007

Corrected and extended by R. J. Mathar, Nov 11 2007

A373241 T(n,k) is the difference between the number of different parts and the number of different multiplicities in the k-th partition of n in graded reverse lexicographic ordering (A080577).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 0, 2, 2, 1, 1, 1, 0, 0, 1, 0, 3, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Olivier Gérard, May 29 2024

This difference is always nonnegative.
The number of zero values in each row is A098859.
The number of ones in each row is A325244.
The number of positive entries in each row is A336866.
The corresponding regular triangle for partitions of n of length k is A373242.
The sum of each row is A373243.

			The array begins
  0
  0,0
  0,1,0
  0,1,0,0,0
  0,1,1,0,0,0,0
  0,1,1,0,0,2,0,0,1,0,0
  0,1,1,0,1,2,0,0,0,1,0,0,0,0,0
  0,1,1,0,1,2,0,0,2,0,1,0,0,1,1,1,0,0,0,0,0,0
  0,1,1,0,1,2,0,1,2,0,1,0,0,2,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,0
  ...

Olivier Gérard, Table of n, a(n) for n = 1..215307

Cf. A373269 a triangle of the same shape and order for number of multiplicities.

Cf. A080577, A098859, A325244, A336866, A373242, A373243.

Flatten @ Table[
  Map[Length[Union[#]] - Length[Union[Length /@ Split[#]]] &,
   IntegerPartitions[n]], {n, 1, 20}]

A373269 T(n,k) is the number of different multiplicities in the k-th partition of n in graded reverse lexicographic ordering (A080577).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 1

Offset: 1

Olivier Gérard, May 29 2024

The regular array for partitions of n of length k is A373270.

Row sums are A373271.

			Array begins:
  1,
  1,1,
  1,1,1,
  1,1,1,2,1,
  1,1,1,2,2,2,1,
  1,1,1,2,1,1,2,1,1,2,1,
  1,1,1,2,1,1,2,2,2,2,2,2,2,2,1
  ...
T(10,34) is the first term with value 3. It corresponds to partition 3+2+2+1+1+1 of 10, which has three different multiplicities.

Olivier Gérard, Table of n, a(n) for n = 1..215307

Flatten@Table[
  Map[Length[Union[Length /@ Split[#]]] &, IntegerPartitions[n]], {n,
   1, 20}]

A176207 Permutations of partitions listed in A080577 with partition lengths listed in A176208; the table has shape A058884.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 2, 3, 1, 4, 1, 3, 1, 1, 2, 2, 1, 2, 1, 1, 2, 4, 2, 3, 1, 1, 5, 1, 4, 1, 1, 3, 2, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 4, 2, 5, 2, 4, 1, 2, 3, 2, 2, 3, 1, 1, 1, 6, 1, 5, 1, 1, 4, 2, 1, 4, 1, 1, 1, 3, 3, 1, 3, 2, 1, 1, 3, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 3

Alford Arnold, Apr 12 2010

The permutations are selected by considering partial sums of A080577:
  1
  1 2 11
  1 2 11 3 21 111
  ...
then prepending values from A176206 yielding
  1
  2 11
  3 21 12 111
  4 31 22 211 13 121 1111
  ...
Cases appearing in A080577 are excluded from {a(n)}.

			Triangle begins:
  {{1,2}},
  {{1,3}, {1,2,1}},
  {{2,3}, {1 4}, {1,3,1}, {1,2,2}, {1,2,1,1}},
Or more concisely:
  {12},
  {13, 121},
  {23, 14, 131, 122, 1211},
  {24, 231, 15, 141, 132, 1311, 1221, 12111},
  ...

Andrew Howroyd, Table of n, a(n) for n = 3..1607 (rows 3..12)

Cf. A058884, A080577, A176206, A176208.

\\ here R(n) returns n-th row as vector of vectors.
L(n,k)={vecsort([Vecrev(p) | p<-partitions(k), p[#p] > n-k], , 4)}
R(n)={ concat(vector(n-1, k, [concat([n-k],p) | p<-L(n,k)])) }
{ for(n=3, 6, print(concat(R(n)))) } \\ Andrew Howroyd, Apr 21 2023

Offset corrected and a(50) and beyond from Andrew Howroyd, Apr 21 2023

A213952 Consider the partitions of n in reverse lexicographic ordering (A080577), a(n) is the position of the partition of n which has the maximum LCM. See A000793.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 5, 8, 15, 13, 33, 49, 35, 49, 73, 107, 143, 211, 293, 398, 505, 510, 685, 710, 948, 740, 994, 2033, 1735, 2266, 1780, 2333, 4653, 5923, 7311, 9213, 7683, 9719, 17878, 14703, 19072, 22814, 28266, 34878, 42876, 52390

Offset: 1

Robert G. Wilson v, Jul 04 2012

nonn

As n grows, a(n)/P(n) -> ~1/3, where P(n) is A000041(n).

			a(5) = 3 because of the seven partitions of 5, {{5}, {4, 1}, {3, 2}, {3, 1, 1}, {2, 2, 1}, {2, 1, 1, 1}, {1, 1, 1, 1, 1}}; the LCMs of each are: {5, 4, 6, 3, 2, 2, 1}. The third one is the maximum.

Robert G. Wilson v, Table of n, a(n) for n = 1..84

Cf. A000793, A080577, A000041.

f[n_] := Block[{lst = Apply[LCM, IntegerPartitions@ n, 1]}, Flatten[ Position[ lst, Max@ lst, 1, 1], 1][[1]]]; Array[f, 47]

A376379 Heinz numbers of integer partitions (x_1, ..., x_k) with at least 2 parts, sorted by increasing multinomial coefficients (x_1 + ... + x_k)!/(x_1! * ... * x_k!). In case of ties, the partitions are sorted in standard order as in A080577.

Original entry on oeis.org

4, 6, 10, 14, 8, 9, 22, 26, 34, 38, 15, 46, 58, 12, 62, 74, 82, 21, 86, 94, 106, 118, 122, 20, 25, 134, 33, 142, 146, 158, 16, 166, 178, 194, 202, 39, 206, 214, 18, 28, 218, 226, 254, 262, 274, 35, 278, 51, 298, 302, 314, 326, 334, 346, 44, 358, 362, 382, 57

Offset: 1

Pontus von Brömssen, Sep 23 2024

nonn

This is a permutation of the composite numbers A002808.

			  n | A376367(n) | partition | a(n)
  --+------------+-----------+-----
  1 |     2      |  (1,1)    |   4
  2 |     3      |  (2,1)    |   6
  3 |     4      |  (3,1)    |  10
  4 |     5      |  (4,1)    |  14
  5 |     6      |  (1,1,1)  |   8
  6 |     6      |  (2,2)    |   9
  7 |     6      |  (5,1)    |  22
The number 210 appears 6 times in A376367, corresponding to the partitions (4,1,1,1), (3,2,2), (6,4), (13,1,1), (19,2), and (209,1), with Heinz numbers 56, 45, 91, 164, 201 and 2578, respectively. These numbers appear as a(257), ..., a(262).

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A002808, A080577, A318762, A376367.

A318762(a(n)) = A376367(n).

A124920 Location of record values in A080577; also partial sums of A006128 plus 1.

Original entry on oeis.org

1, 2, 5, 11, 23, 43, 78, 132, 218, 346, 538, 813, 1212, 1768, 2548, 3616, 5079, 7044, 9688, 13186, 17816, 23868, 31767, 41973, 55147, 71998, 93520, 120814, 155359, 198812, 253375, 321510, 406437, 511803, 642265, 803141, 1001155, 1243967

Offset: 1

Alford Arnold, Nov 13 2006

			A080577 begins
1
2 11
3 21 111
4 31 22 211 1111
5 41 32 311 221 2111 11111
6 51 42 411 33 321 3111 222 2211 21111 111111
therefore A124920 begins 1 2 5 11 23 ...

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000041, A006128, A080577.

A008284 := proc(n,k) if n >= 1 and n = k or k = 1 then 1 elif k > n then 0 else add( A008284(n-k,i),i=1..k) ; fi ; end: A006128 := proc(n) add( k*A008284(n,k),k=1..n) ; end: a := 1 : printf("%d,",a) ; for n from 2 to 80 do a := a + A006128(n-1) : printf("%d,",a) ; od : # R. J. Mathar, Jan 13 2007

A124920(n) = A124920(n-1)+ A006128(n-1), n>1; a(1) = 1.
G.f.: x/(1 - x) + Sum_{i>=1} i*x^(i+1)/(1 - x) * Product_{j=1..i} 1/(1 - x^j). - Ilya Gutkovskiy, Apr 04 2017
a(n) ~ exp(Pi*sqrt(2*n/3)) * (log(6*n) + 2*gamma - 2*log(Pi)) * sqrt(3) / (4*Pi^2), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, May 19 2018

More terms from R. J. Mathar, Jan 13 2007

Clarification of name from Ilya Gutkovskiy, Apr 04 2017

cp's OEIS Frontend

A063008 Canonical partition sequence (see A080577) encoded by prime factorization. The partition [p1,p2,p3,...] with p1 >= p2 >= p3 >= ... is encoded as 2^p1 * 3^p2 * 5^p3 * ... .

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A331581 Maximum part of the n-th integer partition in graded reverse-lexicographic order (A080577); a(1) = 0.

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A138151 Irregular triangle read by rows in which rows 1..n (when read together) list all the parts in the partitions of n and row n starts with the partitions of n that do not contain 1 as a part (in the order used for A080577).

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A131822 Increment each prime factor for each term of the least prime signature sequence derived from A080577.

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A373241 T(n,k) is the difference between the number of different parts and the number of different multiplicities in the k-th partition of n in graded reverse lexicographic ordering (A080577).

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A373269 T(n,k) is the number of different multiplicities in the k-th partition of n in graded reverse lexicographic ordering (A080577).

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A176207 Permutations of partitions listed in A080577 with partition lengths listed in A176208; the table has shape A058884.

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A213952 Consider the partitions of n in reverse lexicographic ordering (A080577), a(n) is the position of the partition of n which has the maximum LCM. See A000793.

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