A181317 -id:A181317 - cp's OEIS frontend

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.

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

A302246 Irregular triangle read by rows in which row n lists all parts of all partitions of n, in nonincreasing order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Apr 05 2018

Also due to the correspondence divisor/part row n lists the terms of the n-th row of A338156 in nonincreasing order. In other words: row n lists in nonincreasing order the divisors of the terms of the n-th row of A176206. - Omar E. Pol, Jun 16 2022

			Triangle begins:
  1;
  2,1,1;
  3,2,1,1,1,1;
  4,3,2,2,2,1,1,1,1,1,1,1;
  5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1;
  6,5,4,4,3,3,3,3,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
  ...
For n = 4 the partitions of 4 are [4], [2, 2], [3, 1], [2, 1, 1], [1, 1, 1, 1]. There is only one 4, only one 3, three 2's and seven 1's, so the 4th row of this triangle is [4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1].
On the other hand for n = 4 the 4th row of A176206 is [4, 3, 2, 2, 1, 1, 1] and the divisors of these terms are [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1], [1] the same as the 4th row of A338156. These divisors listed in nonincreasing order give the 4th row of this triangle. - _Omar E. Pol_, Jun 16 2022

Paolo Xausa, Table of n, a(n) for n = 1..9687, (rows 1..18 of triangle, flattened).

Both column 1 and 2 are A000027.
Row n has length A006128(n).
The sum of row n is A066186(n).
The number of parts k in row n is A066633(n,k).
The sum of all parts k in row n is A138785(n,k).
The number of parts >= k in row n is A181187(n,k).
The sum of all parts >= k in row n is A206561(n,k).
The number of parts <= k in row n is A210947(n,k).
The sum of all parts <= k in row n is A210948(n,k).
First differs from A036037, A080577, A181317, A237982 and A239512 at a(13) = T(4,3).
Cf. A302247 (mirror).
Cf. A000041, A027550, A176206, A221529, A336812, A338156.

nrows=10;Array[ReverseSort[Flatten[IntegerPartitions[#]]]&,nrows] (* Paolo Xausa, Jun 16 2022 *)

row(n) = my(list = List()); forpart(p=n, for (k=1, #p, listput(list, p[k]));); vecsort(Vec(list), , 4); \\ Michel Marcus, Jun 16 2022

A067855 Square of the Euclidean length of the vector of Littlewood-Richardson coefficients of Sum_{lambda |- n} s_lambda^2, where s_lambda are the symmetric Schur functions and the sum runs over all partitions lambda of n.

Original entry on oeis.org

1, 2, 8, 26, 94, 326, 1196, 4358, 16248, 60854, 230184, 874878, 3343614, 12825418, 49368388, 190554410, 737328366, 2858974502, 11106267880, 43215101102, 168398785002, 657070401106, 2566847255572, 10038191414610, 39295007540748

Offset: 0

Richard Stanley, Feb 15 2002

Original name: "Squared length of sum of s_lambda^2, where s_lambda is a Schur function and lambda ranges over all partitions of n."
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/2, g(n) = 4. - Seiichi Manyama, Apr 22 2018
The symbol "|-" means "is a partition of", cf. MathWorld link and the Geloun & Ramgoolam paper. The Littlewood-Richardson coefficients allow a product of two Schur functions to be expressed as a linear combination of Schur functions of the corresponding degree. (The Schur functions symmetric in all n variables correspond to Schur polynomials of partitions extended with 0's to length n.) - M. F. Hasler, Jan 19 2020
See A070933 for similar sums of squares of Littlewood-Richardson coefficients. - M. F. Hasler, Jan 20 2020

			For n=3 the s_lambda^2 summed over all partitions of n and decomposed into a sum of Schur functions yields
    s(6) + 2 s(3,3) + 2 s(4,2) + s(5,1) + 2 s(2,2,2) + 2 s(3,2,1) + s(4,1,1)
    + 2 s(2,2,1,1) + s(3,1,1,1) + s(2,1,1,1,1) + s(1,1,1,1,1,1),
  and the sum of the squares of the coefficients {1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1} gives a(3) = 26.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
J. B. Geloun and S. Ramgoolam, Counting Tensor Model Observables and Branched Covers of the 2-Sphere, arXiv preprint arXiv:1307.6490 [hep-th], 2013.
Eric Weisstein's World of Mathematics, Partition.
Wikipedia, Littlewood-Richardson rule, as of Dec 18 2018.
Wikipedia, Schur polynomial, as of Jan 13 2020.

Cf. A001868.
List of partitions: A036037, A080577, A181317, A330370.
Cf. A070933 (Sum_{lambda,mu,nu} (c^{lambda}_{mu,nu})^2, |mu| = |nu| = n).
Cf. A003040 (maximum number of standard tableaux of the Ferrers diagrams of the partitions of n).

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1,
      binomial(n+n, n), add(b(j, 1)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..33);  # Alois P. Heinz, Aug 24 2019

Table[Tr[(Apply[List,
  Sum[Tr[s @@@ LRRule[\[Lambda], \[Lambda]]],
   {\[Lambda], Partitions[n]}]] /. s[] -> 1)^2], {n, 1, 10}];
(* with 'LRRule' defined in http://users.telenet.be/Wouter.Meeussen/ToolBox.nb - Wouter Meeussen, Jan 19 2020 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 1, Binomial[n+n, n],
     Sum[b[j, 1]*b[n - i*j, i-1], {j, 0, n/i}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Jan 02 2022, after Alois P. Heinz *)

A067855_upto(N)=Vec(1/sqrt(prod(i=1,N-1,1-4*'x^i+O('x^N)))) \\ M. F. Hasler, Jan 23 2020

G.f.: 1/sqrt(Product_{i >= 1} (1 - 4*x^i)).
Euler transform of A001868(n)/2. a(n) = Sum_{pi} Product_{m=1..n} binomial(2*p(m), p(m)), where pi runs through all nonnegative solutions of p(1) + 2*p(2) + ... + n*p(n)=n. - Vladeta Jovovic, Mar 25 2006
a(n) ~ 2^(2*n) / sqrt(c*Pi*n), where c = QPochhammer[1/4] = 0.688537537120339... - Vaclav Kotesovec, Apr 22 2018
By definition, a(n) = Sum_{mu |- 2n} c_mu^2 where Sum_{lambda |- n} s_lambda^2 = Sum_{mu |- 2n} c_mu s_mu, where s_lambda are the Schur polynomials (symmetric in 2n variables) and the sums run over all partitions of n resp. 2n. - M. F. Hasler, Jan 19 2020

More terms from Vladeta Jovovic, Mar 25 2006

Name edited by M. F. Hasler following observations by Wouter Meeussen, Jan 17 2020

A330370 Irregular triangle read by rows T(n,m) in which row n lists all partitions of n ordered by their k-th ranks, or by their k-th largest parts if all their k-th ranks are zeros, with k = 1..n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 12 2019

Theorem: the k-th part of a partition in nonincreasing order of a positive integer equals the number of parts >= k of its conjugate partition.
Example: for n = 9 consider the partition [5, 3, 1]. The first part is 5, so the conjugate partition [3, 2, 2, 1, 1] has five parts >= 1. The second part is 3, so the conjugate partition has three parts >= 2. The third part is 1, so the conjugate partition has only one part >= 3. And vice versa, consider now the partition [3, 2, 2, 1, 1]. The first part is 3, so the conjugate partition [5, 3, 1] has three parts >= 1. The second part is 2, so the conjugate partition has two parts >= 2. The third part is 2, so the conjugate partition has two parts >= 3. The fourth part is 1, so the conjugate partition has only one part >= 4. The fifth part is 1, so the conjugate partition has only one part >= 5.
Corollary: the difference between the k-th part and the (k+1)-st part of a partition in nonincreasing order of a positive integer equals the number of k's in its conjugate partition.
Example: consider the partition [5, 3, 1]. The difference between the first and the second parts is 5 - 3 = 2, which equals the number of 1's in its conjugate partition [3, 2, 2, 1, 1]. The difference between the second and third parts is 3 - 1 = 2, which equals the number of 2's in its conjugate partition. The difference between the third part and the fourth (virtual) part is 1 - 0 = 1, which equals the number of 3's in its conjugate partition. And vice versa, consider the partition [3, 2, 2, 1, 1]. The difference between the first and second parts is 3 - 2 = 1, which equals the number of 1's in its conjugate partition [5, 3, 1]. The difference between the second and third parts is 2 - 2 = 0, which equals the number of 2's in its conjugate partition. The difference between the third and fourth parts is 2 - 1 = 1, which equals the number of 3's in its conjugate partition, and so on.
Self-conjugate partitions are included in all the above comments.
A proof without words is as shown below:
.
                       +------------------------+
                       | +--------------------+ |
                       | | +----------------+ | |
                       | | |                | | |
                       v v v   P2  FD  k    | | |
                                            | | |
           +-------->  * * *   3   1   1  --+ | |
           | +------>  * *     2   0   2      | |
           | +------>  * *     2   1   3  ----+ |
           | | +---->  *       1   0   4        |
           | | +---->  *       1   1   5  ------+
           | | |
           | | |  P1   5 3 1
           | | |
           | | |  FD   2 2 1
           | | |
           | | |  k    1 2 3
           | | |
           | | |       | | |
           | | +-------+ | |
           | +-----------+ |
           +---------------+
.
Every partition of n has n ranks.
The k-th rank of a partition is the k-th part minus the number of parts >= k.
In accordance with the above theorem, the k-th rank of a partition is also the number of parts >= k of its conjugate partition minus the number of parts >= k of the partition.
All ranks of a partition are zeros if and only if the partition is a self-conjugate partition.
The list of ranks of a partition of n equals the list of ranks multipled by -1 of its conjugate partition.
For example, the nine ranks of the partition [5, 3, 1] are [2, 1, -1, -1, -1, -1, 0, 0, 0], and the nine ranks of its conjugate partition [3, 2, 2, 1, 1] are [-2, -1, 1, 1, 1, 1, 0, 0, 0].
Note that the first rank coincides with the Dyson's rank because the first part of a partition is also the largest part, and the number of parts >= 1 is also the total number of parts.
In this triangle the partitions of n appears ordered by their first rank. The partitions that have the same first rank appears ordered by their second rank. The partitions that have the same first rank and the same second rank appears ordered by their third rank, and so on. The partitions that have all k-ranks equal zero appears ordered by their largest parts, then by their second largest parts, then by their third largest parts, and so on.
Note that a partition and its conjugate partition both are equidistants from the center of the list of partitions of n.
The first ranks of the partitions of this triangle give A330368.
For more information about the k-th ranks see A208478.
First differs from A080577 at a(48), and from A036037 at a(56), and from A181317 at a(105).

			Triangle begins:
  [1];
  [2], [1,1];
  [3], [2,1], [1,1,1];
  [4], [3,1], [2,2], [2,1,1], [1,1,1,1];
  [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1];
  [6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [3,1,1,1], [2,2,2], ...
  ...
Illustration of initial terms with a symmetric arrangement (note that the self-conjugate partitions are located in the main diagonal):
.
  1    1 1    1 1 1    1 1 1 1    1 1 1 1 1           1 1 1 1 1 1
  *    * *    * * *    * * * *    * * * * *           * * * * * *
  2
  *
  *
  3           2 1      2 1 1      2 1 1 1             2 1 1 1 1
  *           * *      * * *      * * * *             * * * * *
  *           *        *          *                   *
  *
  4           3 1      2 2        2 2 1               2 2 1 1
  *           * *      * *        * * *               * * * *
  *           *        * *        * *                 * *
  *           *
  *
  5           4 1      3 2        3 1 1               2 2 2
  *           * *      * *        * * *               * * *
  *           *        * *        *                   * * *
  *           *        *          *
  *           *                                       3 1 1 1
  *                                                   * * * *
                                                      *
                                                      *
.
  6           5 1      4 2        3 3      4 1 1      3 2 1
  *           * *      * *        * *      * * *      * * *
  *           *        * *        * *      *          * *
  *           *        *          * *      *          *
  *           *        *                   *
  *           *
  *
For n = 9 the 9th row of the triangle contains the partitions ordered as shown below:
---------------------------------------------------------------------------------
                                                                Ranks
          Conjugate
Label       with        Partitions                k = 1  2  3  4  5  6  7  8  9
---------------------------------------------------------------------------------
   1         30         [9]                           8 -1 -1 -1 -1 -1 -1 -1 -1
   2         29         [8, 1]                        6  0 -1 -1 -1 -1 -1 -1  0
   3         28         [7, 2]                        5  0 -1 -1 -1 -1 -1  0  0
   4         27         [6, 3]                        4  1 -2 -1 -1 -1  0  0  0
   5         26         [7, 1, 1]                     4  0  0 -1 -1 -1 -1  0  0
   6         25         [5, 4]                        3  2 -2 -2 -1  0  0  0  0
   7         24         [6, 2, 1]                     3  0  0 -1 -1 -1  0  0  0
   8         23         [5, 3, 1]                     2  1 -1 -1 -1  0  0  0  0
   9         22         [6, 1, 1, 1]                  2  0  0  0 -1 -1  0  0  0
  10         21         [5, 2, 2]                     2 -1  1 -1 -1  0  0  0  0
  11         20         [4, 4, 1]                     1  2 -1 -2  0  0  0  0  0
  12         19         [5, 2, 1, 1]                  1  0  0  0 -1  0  0  0  0
  13         18         [4, 3, 2]                     1  0  0 -1  0  0  0  0  0
  14         17         [4, 3, 1, 1]                  0  1 -1  0  0  0  0  0  0
  15  (self-conjugate)  [5, 1, 1, 1, 1]  All zeros -> 0  0  0  0  0  0  0  0  0
  16  (self-conjugate)  [3, 3, 3]        All zeros -> 0  0  0  0  0  0  0  0  0
  17         14         [4, 2, 2, 1]                  0 -1  1  0  0  0  0  0  0
  18         13         [3, 3, 2, 1]                 -1  0  0  1  0  0  0  0  0
  19         12         [4, 2, 1, 1, 1]              -1  0  0  0  1  0  0  0  0
  20         11         [3, 2, 2, 2]                 -1 -2  1  2  0  0  0  0  0
  21         10         [3, 3, 1, 1, 1]              -2  1 -1  1  1  0  0  0  0
  22          9         [4, 1, 1, 1, 1, 1]           -2  0  0  0  1  1  0  0  0
  23          8         [3, 2, 2, 1, 1]              -2 -1  1  1  1  0  0  0  0
  24          7         [3, 2, 1, 1, 1, 1]           -3  0  0  1  1  1  0  0  0
  25          6         [2, 2, 2, 2, 1]              -3 -2  2  2  1  0  0  0  0
  26          5         [3, 1, 1, 1, 1, 1, 1]        -4  0  0  1  1  1  1  0  0
  27          4         [2, 2, 2, 1, 1, 1]           -4 -1  2  1  1  1  0  0  0
  28          3         [2, 2, 1, 1, 1, 1, 1]        -5  0  1  1  1  1  1  0  0
  29          2         [2, 1, 1, 1, 1, 1, 1, 1]     -6  0  1  1  1  1  1  1  0
  30          1         [1, 1, 1, 1, 1, 1, 1, 1, 1]  -8  1  1  1  1  1  1  1  1
.
Two examples of the order of partitions:
1) The partitions [6, 3] and [7, 1, 1] both have their first rank equal to 4, so they are ordered by their sencond rank.
2) The self-conjugate partitions [5, 1, 1, 1, 1] and [3, 3, 3] both have all their ranks equal to zero, so they are ordered by their first part.

Row n contains A000041(n) partitions.
Row n has length A006128(n).
The sum of n-th row is A066186(n).
Cf. A000700, A036037, A067619, A080577, A181317, A207031, A330368, A330372, A330373.
For "k-th rank" of a partition see also: A181187, A208478, A208479, A208482, A208483.

A330371 Irregular triangle read by rows T(n,m) in which row n lists all partitions of n ordered by the lower value of their k-th ranks, or by their k-th largest parts if all their k-th ranks are zeros, with k = n..1.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 15 2019

In this triangle the partitions of n are ordered by their n-th rank. The partitions that have the same n-th rank appears ordered by their (n-1)-st rank. The partitions that have the same n-th rank and the same (n-1)-st rank appears ordered by their (n-2)-nd rank, and so on. The partitions that have all k-ranks equal zero appears ordered by their largest parts, then by their second largest parts, then by their third largest parts, and so on.
Note that a partition and its conjugate partition both are equidistants from the center of the list of partitions of n.
For further information see A330370.
First differs from A036037, A181317, A330370 and A334439 at a(48).
First differs from A080577 at a(56).

			Triangle begins:
[1];
[2], [1,1];
[3], [2,1], [1,1,1];
[4], [3,1], [2,2], [2,1,1], [1,1,1,1];
[5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1];
[6], [5,1], [4,2], [4,1,1], [3,3], [3,2,1], [2,2,2], [3,1,1,1], [2,2,1,1], ...
.
For n = 9 the 9th row of the triangle contains the partitions ordered as shown below:
---------------------------------------------------------------------------------
                                                                Ranks
          Conjugate
Label     with label    Partition                 k = 1  2  3  4  5  6  7  8  9
---------------------------------------------------------------------------------
   1         30         [9]                           8 -1 -1 -1 -1 -1 -1 -1 -1
   2         29         [8, 1]                        6  0 -1 -1 -1 -1 -1 -1  0
   3         28         [7, 2]                        5  0 -1 -1 -1 -1 -1  0  0
   4         27         [7, 1, 1]                     4  0  0 -1 -1 -1 -1  0  0
   5         26         [6, 3]                        4  1 -2 -1 -1 -1  0  0  0
   6         25         [6, 2, 1]                     3  0  0 -1 -1 -1  0  0  0
   7         24         [6, 1, 1, 1]                  2  0  0  0 -1 -1  0  0  0
   8         23         [5, 4]                        3  2 -2 -2 -1  0  0  0  0
   9         22         [5, 3, 1]                     2  1 -1 -1 -1  0  0  0  0
  10         21         [5, 2, 2]                     2 -1  1 -1 -1  0  0  0  0
  11         20         [5, 2, 1, 1]                  1  0  0  0 -1  0  0  0  0
  12         19         [4, 4, 1]                     1  2 -1 -2  0  0  0  0  0
  13         18         [4, 3, 2]                     1  0  0 -1  0  0  0  0  0
  14         17         [4, 3, 1, 1]                  0  1 -1  0  0  0  0  0  0
  15  (self-conjugate)  [5, 1, 1, 1, 1]  All zeros -> 0  0  0  0  0  0  0  0  0
  16  (self-conjugate)  [3, 3, 3]        All zeros -> 0  0  0  0  0  0  0  0  0
  17         14         [4, 2, 2, 1]                  0 -1  1  0  0  0  0  0  0
  18         13         [3, 3, 2, 1]                 -1  0  0  1  0  0  0  0  0
  19         12         [3, 2, 2, 2]                 -1 -2  1  2  0  0  0  0  0
  20         11         [4, 2, 1, 1, 1]              -1  0  0  0  1  0  0  0  0
  21         10         [3, 3, 1, 1, 1]              -2  1 -1  1  1  0  0  0  0
  22          9         [3, 2, 2, 1, 1]              -2 -1  1  1  1  0  0  0  0
  23          8         [2, 2, 2, 2, 1]              -3 -2  2  2  1  0  0  0  0
  24          7         [4, 1, 1, 1, 1, 1]           -2  0  0  0  1  1  0  0  0
  25          6         [3, 2, 1, 1, 1, 1]           -3  0  0  1  1  1  0  0  0
  26          5         [2, 2, 2, 1, 1, 1]           -4 -1  2  1  1  1  0  0  0
  27          4         [3, 1, 1, 1, 1, 1, 1]        -4  0  0  1  1  1  1  0  0
  28          3         [2, 2, 1, 1, 1, 1, 1]        -5  0  1  1  1  1  1  0  0
  29          2         [2, 1, 1, 1, 1, 1, 1, 1]     -6  0  1  1  1  1  1  1  0
  30          1         [1, 1, 1, 1, 1, 1, 1, 1, 1]  -8  1  1  1  1  1  1  1  1

Another version of A330370.
Row n contains A000041(n) partitions.
Row n has length A006128(n).
The sum of n-th row is A066186(n).
Cf. A000700, A036037, A080577, A181317, A207031, A330368, A330370, A334439.
For the "k-th rank" see also: A181187, A208478, A208479, A208482, A208483.

cp's OEIS Frontend

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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A302246 Irregular triangle read by rows in which row n lists all parts of all partitions of n, in nonincreasing order.

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A067855 Square of the Euclidean length of the vector of Littlewood-Richardson coefficients of Sum_{lambda |- n} s_lambda^2, where s_lambda are the symmetric Schur functions and the sum runs over all partitions lambda of n.

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A330370 Irregular triangle read by rows T(n,m) in which row n lists all partitions of n ordered by their k-th ranks, or by their k-th largest parts if all their k-th ranks are zeros, with k = 1..n.

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A330371 Irregular triangle read by rows T(n,m) in which row n lists all partitions of n ordered by the lower value of their k-th ranks, or by their k-th largest parts if all their k-th ranks are zeros, with k = n..1.

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