A193173 -id:A193173 - cp's OEIS frontend

A211992 Triangle read by rows in which row n lists the partitions of n in colexicographic order.

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

Offset: 1

Omar E. Pol, Aug 18 2012

The order of the partitions of every integer is reversed with respect to A026792. For example: in A026792 the partitions of 3 are listed as [3], [2, 1], [1, 1, 1], however here the partitions of 3 are listed as [1, 1, 1], [2, 1], [3].
Row n has length A006128(n). Row sums give A066186. Right border gives A000027. The equivalent sequence for compositions (ordered partitions) is A228525. - Omar E. Pol, Aug 24 2013
The representation of the partitions (for fixed n) is as (weakly) decreasing lists of parts, the order between individual partitions (for the same n) is co-lexicographic. The equivalent sequence for partitions as (weakly) increasing lists and lexicographic order is A026791. - Joerg Arndt, Sep 02 2013

			From _Omar E. Pol_, Aug 24 2013: (Start)
Illustration of initial terms:
-----------------------------------------
n      Diagram          Partition
-----------------------------------------
.       _
1      |_|              1;
.       _ _
2      |_| |            1, 1,
2      |_ _|            2;
.       _ _ _
3      |_| | |          1, 1, 1,
3      |_ _| |          2, 1,
3      |_ _ _|          3;
.       _ _ _ _
4      |_| | | |        1, 1, 1, 1,
4      |_ _| | |        2, 1, 1,
4      |_ _ _| |        3, 1,
4      |_ _|   |        2, 2,
4      |_ _ _ _|        4;
.       _ _ _ _ _
5      |_| | | | |      1, 1, 1, 1, 1,
5      |_ _| | | |      2, 1, 1, 1,
5      |_ _ _| | |      3, 1, 1,
5      |_ _|   | |      2, 2, 1,
5      |_ _ _ _| |      4, 1,
5      |_ _ _|   |      3, 2,
5      |_ _ _ _ _|      5;
.       _ _ _ _ _ _
6      |_| | | | | |    1, 1, 1, 1, 1, 1,
6      |_ _| | | | |    2, 1, 1, 1, 1,
6      |_ _ _| | | |    3, 1, 1, 1,
6      |_ _|   | | |    2, 2, 1, 1,
6      |_ _ _ _| | |    4, 1, 1,
6      |_ _ _|   | |    3, 2, 1,
6      |_ _ _ _ _| |    5, 1,
6      |_ _|   |   |    2, 2, 2,
6      |_ _ _ _|   |    4, 2,
6      |_ _ _|     |    3, 3,
6      |_ _ _ _ _ _|    6;
...
Triangle begins:
[1];
[1,1], [2];
[1,1,1], [2,1], [3];
[1,1,1,1], [2,1,1], [3,1], [2,2], [4];
[1,1,1,1,1], [2,1,1,1], [3,1,1], [2,2,1], [4,1], [3,2], [5];
[1,1,1,1,1,1], [2,1,1,1,1], [3,1,1,1], [2,2,1,1], [4,1,1], [3,2,1], [5,1], [2,2,2], [4,2], [3,3], [6];
(End)
From _Gus Wiseman_, May 10 2020: (Start)
The triangle with partitions shown as Heinz numbers (A334437) begins:
    1
    2
    4   3
    8   6   5
   16  12  10   9   7
   32  24  20  18  14  15  11
   64  48  40  36  28  30  22  27  21  25  13
  128  96  80  72  56  60  44  54  42  50  26  45  33  35  17
(End)

Joerg Arndt, Table of n, a(n) for n = 1..10000
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Cf. A026791, A141285, A194446, A228531.
The graded reversed version is A026792.
The length-sensitive refinement is A036037.
The version for reversed partitions is A080576.
Partition lengths are A193173.
Partition maxima are A194546.
Partition minima are A196931.
The version for compositions is A228525.
The Heinz numbers of these partitions are A334437.
Cf. A036036, A080577, A193073, A228100, A296150, A331581, A334301, A334302, A334436, A334439, A334442.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Join@@Table[Sort[IntegerPartitions[n],colex],{n,0,6}] (* Gus Wiseman, May 10 2020 *)

gen_part(n)=
{  /* Generate partitions of n as weakly increasing lists (order is lex): */
    my(ct = 0);
    my(m, pt);
    my(x, y);
    \\ init:
    my( a = vector( n + (n<=1) ) );
    a[1] = 0;  a[2] = n;  m = 2;
    while ( m!=1,
        y = a[m] - 1;
        m -= 1;
        x = a[m] + 1;
        while ( x<=y,
            a[m] = x;
            y = y - x;
            m += 1;
        );
        a[m] = x + y;
        pt = vector(m, j, a[j]);
    /* for A026791 print partition: */
\\        for (j=1, m, print1(pt[j],", ") );
    /* for A211992 print partition as weakly decreasing list (order is colex): */
        forstep (j=m, 1, -1, print1(pt[j],", ") );
        ct += 1;
    );
    return(ct);
}
for(n=1, 10, gen_part(n) );
\\ Joerg Arndt, Sep 02 2013

A026791 Triangle in which n-th row lists juxtaposed lexicographically ordered partitions of n; e.g., the partitions of 3 (1+1+1,1+2,3) appear as 1,1,1,1,2,3 in row 3.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

Differs from A080576 in a(18): Here, (...,1+3,2+2,4), there (...,2+2,1+3,4).
The representation of the partitions (for fixed n) is as (weakly) increasing lists of parts, the order between individual partitions (for the same n) is lexicographic (see example). - Joerg Arndt, Sep 03 2013
The equivalent sequence for compositions (ordered partitions) is A228369. - Omar E. Pol, Oct 19 2019

			First six rows are:
[[1]];
[[1, 1], [2]];
[[1, 1, 1], [1, 2], [3]];
[[1, 1, 1, 1], [1, 1, 2], [1, 3], [2, 2], [4]];
[[1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 1, 3], [1, 2, 2], [1, 4], [2, 3], [5]];
[[1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 2], [1, 1, 1, 3], [1, 1, 2, 2], [1, 1, 4], [1, 2, 3], [1, 5], [2, 2, 2], [2, 4], [3, 3], [6]];
...
From _Omar E. Pol_, Sep 03 2013: (Start)
Illustration of initial terms:
----------------------------------
.                     Ordered
n  j      Diagram     partition j
----------------------------------
.               _
1  1           |_|    1;
.             _ _
2  1         | |_|    1, 1,
2  2         |_ _|    2;
.           _ _ _
3  1       | | |_|    1, 1, 1,
3  2       | |_ _|    1, 2,
3  3       |_ _ _|    3;
.         _ _ _ _
4  1     | | | |_|    1, 1, 1, 1,
4  2     | | |_ _|    1, 1, 2,
4  3     | |_ _ _|    1, 3,
4  4     |   |_ _|    2, 2,
4  5     |_ _ _ _|    4;
...
(End)

Alois P. Heinz, Rows n = 1..19, flattened
Wikiversity, Lexicographic and colexicographic order

Row lengths are given in A006128.
Partition lengths are in A193173.
Other partition orderings: A026792, A036037, A080577, A125106, A139100, A181087, A181317, A182937, A228100, A240837, A242628.
Row lengths are A000041.
Partition sums are A036042.
Partition minima are A196931.
Partition maxima are A194546.
The reflected version is A211992.
The length-sensitive version (sum/length/lex) is A036036.
The colexicographic version (sum/colex) is A080576.
The version for non-reversed partitions is A193073.
Compositions under the same ordering (sum/lex) are A228369.
The reverse-lexicographic version (sum/revlex) is A228531.
The Heinz numbers of these partitions are A334437.
Cf. A049085, A103921, A112798, A115623, A129129, A331581, A334435, A334439, A334442.

T:= proc(n) local b, ll;
      b:= proc(n,l)
            if n=0 then ll:= ll, l[]
          else seq(b(n-i, [l[], i]), i=`if`(l=[],1,l[-1])..n)
            fi
          end;
      ll:= NULL; b(n, []); ll
    end:
seq(T(n), n=1..8);  # Alois P. Heinz, Jul 16 2011

T[n0_] := Module[{b, ll}, b[n_, l_] := If[n == 0, ll = Join[ll, l], Table[ b[n - i, Append[l, i]], {i, If[l == {}, 1, l[[-1]]], n}]]; ll = {}; b[n0, {}]; ll]; Table[T[n], {n, 1, 8}] // Flatten (* Jean-François Alcover, Aug 05 2015, after Alois P. Heinz *)
Table[DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions[n]], x_ /; x == 0, 2], {n, 7}] // Flatten (* Robert Price, May 18 2020 *)

t = [[[]]]
for n in range(1, 10):
    p = []
    for minp in range(1, n):
        p += [[minp] + pp for pp in t[n-minp] if min(pp) >= minp]
    t.append(p + [[n]])
print(t)
# Andrey Zabolotskiy, Oct 18 2019

A334437 Heinz number of the n-th reversed integer partition in graded lexicographical order.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 5, 16, 12, 10, 9, 7, 32, 24, 20, 18, 14, 15, 11, 64, 48, 40, 36, 28, 30, 22, 27, 21, 25, 13, 128, 96, 80, 72, 56, 60, 44, 54, 42, 50, 26, 45, 33, 35, 17, 256, 192, 160, 144, 112, 120, 88, 108, 84, 100, 52, 90, 66, 70, 34, 81, 63, 75, 39, 55, 49, 19

Offset: 0

Gus Wiseman, May 03 2020

A permutation of the positive integers.
Reversed integer partitions are finite weakly increasing sequences of positive integers. The non-reversed version is A334434.
This is the graded reverse of the so-called "Mathematica" order (A080577, A129129).
The Heinz number of a reversed integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and reversed partitions.
Also Heinz numbers of partitions in colexicographic order (cf. A211992).
As a triangle with row lengths A000041, the sequence starts {{1},{2},{4,3},{8,6,5},...}, so offset is 0.

			The sequence of terms together with their prime indices begins:
    1: {}              11: {5}                 44: {1,1,5}
    2: {1}             64: {1,1,1,1,1,1}       54: {1,2,2,2}
    4: {1,1}           48: {1,1,1,1,2}         42: {1,2,4}
    3: {2}             40: {1,1,1,3}           50: {1,3,3}
    8: {1,1,1}         36: {1,1,2,2}           26: {1,6}
    6: {1,2}           28: {1,1,4}             45: {2,2,3}
    5: {3}             30: {1,2,3}             33: {2,5}
   16: {1,1,1,1}       22: {1,5}               35: {3,4}
   12: {1,1,2}         27: {2,2,2}             17: {7}
   10: {1,3}           21: {2,4}              256: {1,1,1,1,1,1,1,1}
    9: {2,2}           25: {3,3}              192: {1,1,1,1,1,1,2}
    7: {4}             13: {6}                160: {1,1,1,1,1,3}
   32: {1,1,1,1,1}    128: {1,1,1,1,1,1,1}    144: {1,1,1,1,2,2}
   24: {1,1,1,2}       96: {1,1,1,1,1,2}      112: {1,1,1,1,4}
   20: {1,1,3}         80: {1,1,1,1,3}        120: {1,1,1,2,3}
   18: {1,2,2}         72: {1,1,1,2,2}         88: {1,1,1,5}
   14: {1,4}           56: {1,1,1,4}          108: {1,1,2,2,2}
   15: {2,3}           60: {1,1,2,3}           84: {1,1,2,4}
Triangle begins:
    1
    2
    4   3
    8   6   5
   16  12  10   9   7
   32  24  20  18  14  15  11
   64  48  40  36  28  30  22  27  21  25  13
  128  96  80  72  56  60  44  54  42  50  26  45  33  35  17
This corresponds to the following tetrangle:
                  0
                 (1)
               (11)(2)
             (111)(12)(3)
        (1111)(112)(13)(22)(4)
  (11111)(1112)(113)(122)(14)(23)(5)

Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
The constructive version is A026791 (triangle).
The length-sensitive version is A185974.
Compositions under the same order are A228351 (triangle).
The version for non-reversed partitions is A334434.
The dual version (sum/revlex) is A334436.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colexicographic order (sum/length/colex) are A036037.
Graded reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Graded lexicographically ordered partitions are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Graded Heinz numbers are given by A215366.
Sorting partitions by Heinz number gives A296150.
Partitions in dual Abramowitz-Stegun (sum/length/revlex) order are A334439.
Cf. A056239, A066099, A129129, A228531, A333219, A333220, A334301, A334302, A334433, A334435, A334438.

lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
Table[Times@@Prime/@#&/@Sort[Sort/@IntegerPartitions[n],lexsort],{n,0,8}]

A001222(a(n)) = A193173(n).

A238966 The number of distinct primes in divisor lattice in canonical order.

Original entry on oeis.org

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

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

After a(0) = 0, this appears to be the same as A128628. - Gus Wiseman, May 24 2020

Also the number of parts in the n-th integer partition in graded reverse-lexicographic order (A080577). - Gus Wiseman, May 24 2020

			Triangle T(n,k) begins:
  0;
  1;
  1, 2;
  1, 2, 3;
  1, 2, 2, 3, 4;
  1, 2, 2, 3, 3, 4, 5;
  1, 2, 2, 3, 2, 3, 4, 3, 4, 5, 6;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
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.
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row sums are A006128.
Cf. A036043 in canonical order.
Cf. A001221, A063008.
Row lengths are A000041.
The generalization to compositions is A000120.
The sum of the partition is A036042.
The lexicographic version (sum/lex) is A049085.
Partition lengths of A080577.
The partition has A115623 distinct elements.
The Heinz number of the partition is A129129.
The colexicographic version (sum/colex) is A193173.
The maximum of the partition is A331581.
Partitions in lexicographic order (sum/lex) are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Cf. A026792, A036036, A080576, A103921, A112798, A182715, A333486, A334302, A334435, A334436, A334442.

o:= proc(n) option remember; nops(ifactors(n)[2]) end:
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-> o(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Mar 26 2020

revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];
Table[Length/@Sort[IntegerPartitions[n],revlexsort],{n,0,8}] (* Gus Wiseman, May 24 2020 *)
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]]];
P[n_] := P[n] = Product[Prime[i]^#[[i]], {i, 1, Length[#]}]& /@ b[n, n];
T[n_, k_] := PrimeNu[P[n][[k + 1]]];
Table[T[n, k], {n, 0, 9}, {k, 0, Length[P[n]] - 1}] // Flatten (* Jean-François Alcover, Jan 03 2022, after Alois P. Heinz in A063008 *)

Row(n)={apply(s->#s, vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 25 2020

T(n,k) = A001221(A063008(n,k)). - Andrew Howroyd, Mar 25 2020

a(n) = A001222(A129129(n)). - Gus Wiseman, May 24 2020

Offset changed and terms a(50) and beyond from Andrew Howroyd, Mar 25 2020

A194547 Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n, with partitions in lexicographic order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 10 2011

Row n has length A000041(n). The sum of row n is equal to zero.

			Written as a triangle:
  0;
  -1,1;
  -2,0,2;
  -3,-1,1,0,3;
  -4,-2,0,-1,2,1,4;
  -5,-3,-1,-2,1,0,3,-1,2,1,5;
  -6,-4,-2,-3,0,-1,2,-2,1,0,4,0,3,2,6;
  -7,-5,-3,-4,-1,-2,1,-3,0,-1,3,-1,2,1,5,-2,1,0,4,3,2,7;

Alois P. Heinz, Rows n = 1..26, flattened

Cf. A105805, A135010, A138121, A194546, A194548, A194549.

T:= proc(n) local b, l;
      b:= proc(n, i, t)
            if n=0 then l:=l, i-t
          elif i>n then
          else b(n-i, i, t+1); b(n, i+1, t)
            fi
          end;
      l:= NULL; b(n, 1, 0); l
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Dec 22 2011

T[n_] := Module[{b, l}, b[n0_, i_, t_] := If [n0==0, l = Append[l, i-t], If[i>n0, , b[n0-i, i, t+1]; b[n0, i+1, t]]]; l = {}; b[n, 1, 0]; l];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)

a(n) = A194546(n) - A193173(n).

More terms from Alois P. Heinz, Dec 22 2011

A194548 Triangle read by rows: T(n,k) = number of parts in the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 4, 3, 3, 2, 2, 2, 1, 4, 3, 3, 2, 3, 2, 2, 1, 5, 4, 4, 3, 3, 3, 2, 3, 2, 2, 2, 1, 5, 4, 4, 3, 4, 3, 3, 2, 3, 3, 2, 2, 2, 1, 6, 5, 5, 4, 4, 4, 3, 4, 3, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 2, 1, 6, 5, 5, 4, 5, 4, 4, 3, 4, 4, 3, 3, 3, 2, 4, 3, 3, 3, 2, 3, 2, 2, 2, 1

Offset: 1

Omar E. Pol, Dec 11 2011

			Written as a triangle:
  0;
  1;
  1;
  2,1;
  2,1;
  3,2,2,1;
  3,2,2,1;
  4,3,3,2,2,2,1;
  4,3,3,2,3,2,2,1;
  5,4,4,3,3,3,2,3,2,2,2,1;
  5,4,4,3,4,3,3,2,3,3,2,2,2,1;
  6,5,5,4,4,4,3,4,3,3,3,2,4,3,3,2,3,2,2,2,1;
  6,5,5,4,5,4,4,3,4,4,3,3,3,2,4,3,3,3,2,3,2,2,2,1;

Alois P. Heinz, Rows n = 1..33, flattened
Tilman Piesk, Table for A194602, showing the non-one addends.

Row sums give A138135. Row n has length A187219(n).

Cf. A002865, A135010, A138121, A193173, A194546, A194547, A194549.

T:= proc(n) local b, l;
      b:= proc(n, i, t)
            if n=0 then l:=l, t
          elif i>n then
          else b(n-i, i, t+1); b(n, i+1, t)
            fi
          end;
      if n<2 then 0 else l:= NULL; b(n, 2, 0); l fi
    end:
seq(T(n), n=1..15);  # Alois P. Heinz, Dec 19 2011

T[n_] := Module[{b, l}, b[n0_, i_, t_] :=
     If[n0==0, l = Append[l, t],
     If[i>n0, , b[n0-i, i, t+1]; b[n0, i+1, t]]];
     If[n<2, {0}, l = {}; b[n, 2, 0]; l]];
Table[T[n], {n, 1, 15}]  // Flatten (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)

More terms from Alois P. Heinz, Dec 19 2011

A333486 Length of the n-th reversed integer partition in graded reverse-lexicographic order. Partition lengths of A228531.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 23 2020

			Triangle begins:
  0
  1
  1 2
  1 2 3
  1 2 2 3 4
  1 2 2 3 3 4 5
  1 2 2 3 2 3 3 4 4 5 6
  1 2 2 3 2 3 3 4 3 4 4 5 5 6 7
  1 2 2 2 3 3 4 2 3 3 4 3 4 4 5 4 5 5 6 6 7 8

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

Row lengths are A000041.
The generalization to compositions is A000120.
Row sums are A006128.
The same partition has sum A036042.
The length-sensitive version (sum/length/revlex) is A036043.
The colexicographic version (sum/colex) is A049085.
The same partition has minimum A182715.
The lexicographic version (sum/lex) is A193173.
The tetrangle of these partitions is A228531.
The version for non-reversed partitions is A238966.
The same partition has Heinz number A334436.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
Partitions in lexicographic order (sum/lex) are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Partitions in opposite Abramowitz-Stegun order (sum/length/revlex) are A334439.
Cf. A026792, A049085, A080576, A080577, A103921, A112798, A115623, A129129, A331581, A334302, A334435, A334442.

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

A194546 Triangle read by rows: T(n,k) is the largest part of the k-th partition of n, with partitions in colexicographic order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 10 2011

Row n lists the first A000041(n) terms of A141285.

The representation of the partitions (for fixed n) is as (weakly) decreasing lists of parts, the order between individual partitions (for the same n) is co-lexicographic, see example. - Joerg Arndt, Sep 13 2013

			For n = 5 the partitions of 5 in colexicographic order are:
  1+1+1+1+1
  2+1+1+1
  3+1+1
  2+2+1
  4+1
  3+2
  5
so the fifth row is the largest in each partition: 1,2,3,2,4,3,5
Triangle begins:
  1;
  1,2;
  1,2,3;
  1,2,3,2,4;
  1,2,3,2,4,3,5;
  1,2,3,2,4,3,5,2,4,3,6;
  1,2,3,2,4,3,5,2,4,3,6,3,5,4,7;
  1,2,3,2,4,3,5,2,4,3,6,3,5,4,7,2,4,3,6,5,4,8;
...

Wikiversity, Lexicographic and colexicographic order

The sum of row n is A006128(n).
Cf. A135010, A138121, A141285, A194547, A194548, A194549.
Row lengths are A000041.
Let y be the n-th integer partition in colexicographic order (A211992):
- The maximum of y is a(n).
- The length of y is A193173(n).
- The minimum of y is A196931(n).
- The Heinz number of y is A334437(n).
Lexicographically ordered reversed partitions are A026791.
Reverse-colexicographically ordered partitions are A026792.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
Reverse-lexicographically ordered partitions are A080577.
Lexicographically ordered partitions are A193073.
Cf. A036037, A063008, A115623, A228100, A228531, A238966, A334301.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Max/@Join@@Table[Sort[IntegerPartitions[n],colex],{n,8}] (* Gus Wiseman, May 31 2020 *)

a(n) = A061395(A334437(n)). - Gus Wiseman, May 31 2020

Definition corrected by Omar E. Pol, Sep 12 2013

A207034 Sum of all parts minus the number of parts of the n-th partition in the list of colexicographically ordered partitions of j, if 1<=n<=A000041(j).

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 6, 6, 7, 5, 6, 6, 7, 6, 7, 7, 8, 5, 6, 6, 7, 7, 7, 8, 7, 8, 8, 8, 9, 6, 7, 7, 8, 7, 8, 8, 9, 8, 8, 9, 9, 9, 10, 6, 7, 7, 8, 8, 8, 9, 8, 9, 9, 9, 10, 8, 9, 9, 10, 9, 10, 10, 10, 11, 7, 8, 8, 9, 8, 9

Offset: 1

Omar E. Pol, Feb 20 2012

a(n) is also the column number in which is located the part of size 1 in the n-th zone of the tail of the last section of the set of partitions of k in colexicographic order, minus the column number in which is located the part of size 1 in the first row of the same tail, when k -> infinity (see example). For the definition of "section" see A135010.

			Illustration of initial terms, n = 1..15. Consider the last 15 rows of the tail of the last section of the set of partitions in colexicographic order of any integer >= 8. The tail contains at least A000041(8-1) = 15 parts of size 1. a(n) is also the number of dots in the n-th row of the diagram.
----------------------------------
n      Tail                  a(n)
----------------------------------
15        1 . . . . . .       6
14          1 . . . . .       5
13          1 . . . . .       5
12            1 . . . .       4
11          1 . . . . .       5
10            1 . . . .       4
9             1 . . . .       4
8               1 . . .       3
7             1 . . . .       4
6               1 . . .       3
5               1 . . .       3
4                 1 . .       2
3                 1 . .       2
2                   1 .       1
1                     1       0
----------------------------------
Written as a triangle:
0;
1;
2;
2,3;
3,4;
3,4,4,5;
4,5,5,6;
4,5,5,6,6,6,7;
5,6,6,7,6,7,7,8;
5,6,6,7,7,7,8,7,8,8,8,9;
6,7,7,8,7,8,8,9,8,8,9,9,9,10;
6,7,7,8,8,8,9,8,9,9,9,10,8,9,9,10,9,10,10,10,11;
...
Consider a matrix [j X A000041(j)] in which the rows represent the partitions of j in colexicographic order (see A211992). Every part of every partition is located in a cell of the matrix. We can see that a(n) is the number of empty cells in row n for any integer j, if A000041(j) >= n. The number of empty cells in row n equals the sum of all parts minus the number of parts in the n-th partition of j.
Illustration of initial terms. The smallest part of every partition is located in the last column of the matrix.
---------------------------------------------------------
.   j: 1    2       3         4           5             6
n a(n)
---------------------------------------------------------
1  0 | 1  1 1   1 1 1   1 1 1 1   1 1 1 1 1   1 1 1 1 1 1
2  1 |    . 2   . 2 1   . 2 1 1   . 2 1 1 1   . 2 1 1 1 1
3  2 |          . . 3   . . 3 1   . . 3 1 1   . . 3 1 1 1
4  2 |                  . . 2 2   . . 2 2 1   . . 2 2 1 1
5  3 |                  . . . 4   . . . 4 1   . . . 4 1 1
6  3 |                            . . . 3 2   . . . 3 2 1
7  4 |                            . . . . 5   . . . . 5 1
8  3 |                                        . . . 2 2 2
9  4 |                                        . . . . 4 2
10 4 |                                        . . . . 3 3
11 5 |                                        . . . . . 6
...
Illustration of initial terms. In this case the largest part of every partition is located in the first column of the matrix.
---------------------------------------------------------
.   j: 1    2       3         4           5             6
n a(n)
---------------------------------------------------------
1  0 | 1  1 1   1 1 1   1 1 1 1   1 1 1 1 1   1 1 1 1 1 1
2  1 |    2 .   2 1 .   2 1 1 .   2 1 1 1 .   2 1 1 1 1 .
3  2 |          3 . .   3 1 . .   3 1 1 . .   3 1 1 1 . .
4  2 |                  2 2 . .   2 2 1 . .   2 2 1 1 . .
5  3 |                  4 . . .   4 1 . . .   4 1 1 . . .
6  3 |                            3 2 . . .   3 2 1 . . .
7  4 |                            5 . . . .   5 1 . . . .
8  3 |                                        2 2 2 . . .
9  4 |                                        4 2 . . . .
10 4 |                                        3 3 . . . .
11 5 |                                        6 . . . . .
...

Alois P. Heinz, Table of n, a(n) for n = 1..10143

Row r has length A187219(r). Partial sums give A207038. Row sums give A207035. Right border gives A001477. Where records occur give A000041 without repetitions.

Cf. A135010, A138121, A141285, A182703, A194548, A196087, A207031, A207032, A207035, A211992, A228716, A230440.

a(n) = t(n) - A194548(n), if n >= 2, where t(n) is the n-th element of the following sequence: triangle read by rows in which row n lists n repeated k times, where k = A187219(n).
a(n) = A000120(A194602(n-1)) = A000120(A228354(n)-1).
a(n) = i - A193173(i,n), i >= 1, 1<=n<=A000041(i).

A182284 Triangle read by rows: T(n,k) = number of parts in the k-th zone of the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 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, 1

Offset: 1

Omar E. Pol, Apr 23 2012

			Illustration of three arrangements of the last section of the set of partitions of 7 and the zone numbers:
--------------------------------------------------------
Zone \   a)                    b)                    c)
--------------------------------------------------------
15      (7)                   (7)       (. . . . . . 7)
14      (4+3)               (4+3)       (. . . 4 . . 3)
13      (5+2)               (5+2)       (. . . . 5 . 2)
12      (3+2+2)           (3+2+2)       (. . 3 . 2 . 2)
11        (1)                 (1)                   (1)
10          (1)               (1)                   (1)
9           (1)               (1)                   (1)
8             (1)             (1)                   (1)
7           (1)               (1)                   (1)
6             (1)             (1)                   (1)
5             (1)             (1)                   (1)
4               (1)           (1)                   (1)
3               (1)           (1)                   (1)
2                 (1)         (1)                   (1)
1                   (1)       (1)                   (1)
.
For n = 7 and k = 12 we can see that in the 12th zone of the last section there are three parts: 3, 2, 2, therefore T(7,12) = 3.
Written as a triangle begins:
1;
1,1;
1,1,1;
1,1,1,2,1;
1,1,1,1,1,2,1;
1,1,1,1,1,1,1,3,2,2,1;
1,1,1,1,1,1,1,1,1,1,1,3,2,2,1;
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,3,3,2,2,2,1;

Row n has length A000041(n). Row sums give A138137.

Cf. A135010, A138121, A193173, A182285.

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A211992 Triangle read by rows in which row n lists the partitions of n in colexicographic order.

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A026791 Triangle in which n-th row lists juxtaposed lexicographically ordered partitions of n; e.g., the partitions of 3 (1+1+1,1+2,3) appear as 1,1,1,1,2,3 in row 3.

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A334437 Heinz number of the n-th reversed integer partition in graded lexicographical order.

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A238966 The number of distinct primes in divisor lattice in canonical order.

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A194547 Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n, with partitions in lexicographic order.

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A194548 Triangle read by rows: T(n,k) = number of parts in the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.

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A333486 Length of the n-th reversed integer partition in graded reverse-lexicographic order. Partition lengths of A228531.

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