A080577 -id:A080577 - cp's OEIS frontend

A241918 Table of partitions where the ordering is based on the modified partial sums of the exponents of primes in the prime factorization of n.

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

Offset: 1

Antti Karttunen, May 03 2014, based on Marc LeBrun's Jan 11 2006 message on SeqFan mailing list

a(1) = 0 by convention (stands for an empty partition).

For n >= 2, A203623(n-1)+2 gives the index to the beginning of row n and for n>=1, A203623(n)+1 is the index to the end of row n.

			Table begins:
Row     Partition
[ 1]    0;         (stands for empty partition)
[ 2]    1;         (as 2 = 2^1)
[ 3]    1,1;       (as 3 = 2^0 * 3^1)
[ 4]    2;         (as 4 = 2^2)
[ 5]    1,1,1;     (as 5 = 2^0 * 3^0 * 5^1)
[ 6]    2,2;       (as 6 = 2^1 * 3^1)
[ 7]    1,1,1,1;   (as 7 = 2^0 * 3^0 * 5^0 * 7^1)
[ 8]    3;         (as 8 = 2^3)
[ 9]    1,2;       (as 9 = 2^0 * 3^2)
[10]    2,2,2;     (as 10 = 2^1 * 3^0 * 5^1)
[11]    1,1,1,1,1;
[12]    3,3;
[13]    1,1,1,1,1,1;
[14]    2,2,2,2;
[15]    1,2,2;     (as 15 = 2^0 * 3^1 * 5^1)
[16]    4;
[17]    1,1,1,1,1,1,1;
[18]    2,3;       (as 18 = 2^1 * 3^2)
etc.
If n is 2^k (k>=1), then the partition is a singleton {k}, otherwise, add one to the exponent of 2 (= A007814(n)), and subtract one from the exponent of the greatest prime dividing n (= A071178(n)), leaving the intermediate exponents as they are, and then take partial sums of all, thus resulting for e.g. 15 = 2^0 * 3^1 * 5^1 the modified sequence of exponents {0+1, 1, 1-1} -> {1,1,0}, whose partial sums {1,1+1,1+1+0} -> {1,2,2} give the corresponding partition at row 15.

Antti Karttunen, Table of n, a(n) for n = 1..10081; rows 1..521 flattened.
Marc LeBrun's original "crazy order" mapping for partitions (Copy of Marc's Jan 11 2006 message in OEIS Wiki)

For n>=2, the length of row n is given by A061395(n).
Cf. also A067255, A203623, A241914.
Other tables of partitions: A112798 (also based on prime factorization), A227739, A242628 (encoded in the binary representation of n), and A036036-A036037, A080576-A080577, A193073 for various lexicographical orderings.
Permutation A241909 maps between order of partitions employed here, and the order employed in A112798.
Permutation A122111 is induced when partitions in this list are conjugated.
A241912 gives the row numbers for which the corresponding rows in A112798 and here are the conjugate partitions of each other.

Table[If[n == 1, {0}, Function[s, Function[t, Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, s]]]]@ ConstantArray[0, Transpose[s][[1, -1]]]][FactorInteger[n] /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]], {n, 31}] // Flatten (* Michael De Vlieger, May 12 2017 *)

If A241914(n)=0 and A241914(n+1)=0, a(n) = A067255(n); otherwise, if A241914(n)=0 and A241914(n+1)>0, a(n) = A067255(n)+1; otherwise, if A241914(n)>0 and A241914(n+1)=0, a(n) = a(n-1) + A067255(n) - 1, otherwise, when A241914(n)>0 and A241914(n+1)>0, a(n) = a(n-1) + A067255(n).

A334440 Irregular triangle T(n,k) read by rows: row n lists numbers of distinct parts of the n-th integer partition in Abramowitz-Stegun (sum/length/lex) order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 05 2020

The total number of parts, counting duplicates, is A036043. The version for reversed partitions is A103921.

			Triangle begins:
  0
  1
  1 1
  1 2 1
  1 1 2 2 1
  1 2 2 2 2 2 1
  1 1 2 2 1 3 2 2 2 2 1
  1 2 2 2 2 2 3 2 2 3 2 2 2 2 1
  1 1 2 2 2 2 2 3 3 2 1 3 2 3 2 2 3 2 2 2 2 1

Robert Price, Table of n, a(n) for n = 0..9295 (25 rows).
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
The number of not necessarily distinct parts is A036043.
The version for reversed partitions is A103921.
Ignoring length (sum/lex) gives A103921 (also).
a(n) is the number of distinct elements in row n of A334301.
The maximum part of the same partition is A334441.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colex order (sum/length/colex) are A036037.
Graded reverse-lexicographically ordered partitions are A080577.
Partitions counted by sum and number of distinct parts are A116608.
Graded lexicographically ordered partitions are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Partitions in dual Abramowitz-Stegun (sum/length/revlex) order are A334439.
Cf. A001221, A049085, A124734, A185974, A296774, A299755, A334028, A334302, A334433, A334434, A334435, A334438.

Join@@Table[Length/@Union/@Sort[IntegerPartitions[n]],{n,0,10}]

a(n) = A001221(A334433(n)).

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

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

A115729 Number of subpartitions of partitions in Mathematica order.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 4, 5, 7, 6, 7, 5, 6, 9, 9, 10, 9, 9, 6, 7, 11, 12, 10, 13, 14, 10, 13, 12, 11, 7, 8, 13, 15, 14, 16, 19, 16, 16, 17, 19, 14, 16, 15, 13, 8, 9, 15, 18, 18, 15, 19, 24, 23, 22, 19, 21, 26, 22, 23, 15, 21, 24, 18, 19, 18, 15, 9, 10, 17, 21, 22, 20, 22

Offset: 0

Franklin T. Adams-Watters, Mar 11 2006

nonn

subpart([n^k]) = C(n+k,k); subpart([n,n-1,n-2,...,1]) = C_n = A000108(n).

			Partition 5 in Mathematica order is [2,1]; it has 5
subpartitions: [], [1], [2], [1^2] and [2,1] itself.

Cf. A115728, A080577, A000108, A007318.

/* Expects input as vector in decreasing order - e.g. [3,2,1,1] */ subpart2(p)=local(i,j,v,n,k);n=matsize(p)[2];if(n==0,1,v=vector(p[1]+1,i, 1);for(i=1,n,k=p[i];for(j=1,k,v[k+1-j]+=v[k+2-j]));v[1])

For a partition P = [p_1,...,p_n] with the p_i in decreasing order, define b(i,j) to be the number of subpartitions of [p_1,...,p_i] with the i-th part = j (b(i,0) is subpartitions with less than i parts). Then b(1,j)=1 for j<=p_1, b(i+1,j) = Sum_{k=j}^{p_i} b(i,k) for 0<=k<=p_{i+1}; and the total number of subpartitions is sum_{k=1}^{p_n} b(n,k).

A124832 Table of exponents of prime factorizations in A025487.

Original entry on oeis.org

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

Offset: 2

Franklin T. Adams-Watters, Nov 09 2006

This is an enumeration of all partitions.

			From _M. F. Hasler_, Oct 12 2018: (Start)
The table starts as follows:
  n : signature   (A025487(n) = factorization)
  1 : []          (1 = empty product)
  2 : [1]         (2 = 2^1)
  3 : [2]         (4 = 2^2)
  4 : [1, 1]      (6 = 2^1 * 3^1)
  5 : [3]         (8 = 2^3)
  6 : [2, 1]      (12 = 2^2 * 3^1)
  7 : [4]         (16 = 2^4)
  8 : [3,1]       (24 = 2^3 * 3^1)
  9 : [1, 1, 1]   (30 = 2^1 * 3^1 * 5^1)
  etc. (End)

Michael De Vlieger, Table of n, a(n) for n = 2..10820 (rows 2 <= n <= 2500, first 106 terms from Ray Chandler).

Cf. A025487, A036041 (row sums), A061394 (row lengths), A124829, A036036, A080577.

Map[FactorInteger[#][[All, -1]] &, Import["https://oeis.org/A025487/b025487.txt", "Data"][[2 ;; 48, -1]] ] // Flatten (* Michael De Vlieger, Feb 06 2020 *)

A124832_row(n)=factor(A025487(n))[,2] \\ M. F. Hasler, Oct 12 2018

A025487(n) = Product_{k=1..A061394(n)} prime(k)^T(n,k). [Edited by M. F. Hasler, Oct 12 2018]

Erroneous explanations in cross-references corrected by M. F. Hasler, Oct 12 2018

A193173 Triangle in which n-th row lists the number of elements in lexicographically ordered partitions of n, A026791.

Original entry on oeis.org

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

Offset: 1

Alois P. Heinz, Jul 17 2011

This sequence first differs from A049085 in the partitions of 6 (at flattened index 22):
6, 5, 4, 4, 3, 3, 2, 3, 2, 2, 1 (this sequence);
6, 5, 4, 3, 4, 3, 2, 3, 2, 2, 1 (A049085).
- Jason Kimberley, Oct 27 2011
Rows sums give A006128, n >= 1. - Omar E. Pol, Dec 06 2011
The name is correct if the partitions are read in reverse, so that the parts are weakly increasing. The version for non-reversed partitions is A049085.

			The lexicographically ordered partitions of 3 are [[1, 1, 1], [1, 2], [3]], thus row 3 has 3, 2, 1.
Triangle begins:
  1;
  2, 1;
  3, 2, 1;
  4, 3, 2, 2, 1;
  5, 4, 3, 3, 2, 2, 1;
  6, 5, 4, 4, 3, 3, 2, 3, 2, 2, 1;
  ...

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

Row lengths are A000041.
Partition lengths of A026791.
The version ignoring length is A036043.
The version for non-reversed partitions is A049085.
The maxima of these partitions are A194546.
Reversed partitions in Abramowitz-Stegun order are A036036.
Reverse-lexicographically ordered partitions are A080577.
Cf. A001222, A115623, A129129, A185974, A193073, A211992, A228531, A334302, A334434, A334437, A334440, A334441.

T:= proc(n) local b, ll;
      b:= proc(n,l)
            if n=0 then ll:= ll, nops(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..11);

lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
Table[Length/@Sort[Reverse/@IntegerPartitions[n],lexsort],{n,0,10}] (* Gus Wiseman, May 22 2020 *)

A128628 An irregular triangular array read by rows, with shape sequence A000041(n) related to sequence A060850.

Original entry on oeis.org

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: 1

Alford Arnold, Mar 27 2007, Aug 01 2007

The next level gets created from each node by adding one or two more nodes. If a single node is added, its value is one more than the value of its parent. If two nodes are added, the first is equal in value to the parent and the value of the second is one more than the value of the parent.
Sequence A036043 counts the parts of numeric partitions and contains the same values on each row as the current sequence. When a node generates two branches the first branch can be mapped to cyclic partitions; all other branches map to matching partitions.
Appears to be the triangle in which the n-th row contains the number of parts of each partition of n, where the partitions are ordered as in A080577. - Jason Kimberley, May 12 2010

			The values at level three are 1, 2, and 3.
The 1 generates 1 and 2; the 2 generates 2 and 3; the 3 only generates 4.
The array begins
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

T. D. Noe, Rows n = 1..25 of irregular triangle, flattened

Cf. A000041, A060850, A128629.
Cf. A006128 (row sums), A036043.
Cf. A177740.
Cf. A308355 (limiting row sequence).

Flatten[Table[Length /@ IntegerPartitions[n], {n, 9}]] (* T. D. Noe, Feb 27 2014 *)

A344086 Flattened tetrangle of strict integer partitions sorted first by sum, then lexicographically.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 11 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (21)(3)
  4: (31)(4)
  5: (32)(41)(5)
  6: (321)(42)(51)(6)
  7: (421)(43)(52)(61)(7)
  8: (431)(521)(53)(62)(71)(8)
  9: (432)(531)(54)(621)(63)(72)(81)(9)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724.
Triangle sums are A066189.
Taking revlex instead of lex gives A118457.
The not necessarily strict version is A193073.
The version for reversed partitions is A246688.
The Heinz numbers of these partitions grouped by sum are A246867.
The ordered generalization is A339351.
Taking colex instead of lex gives A344087.
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts reversed strict partitions by Heinz number.
A329631 sorts strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080577, A112798, A124734, A162247, A211992, A228100, A228351, A228531, A272020, A299755, A296774, A304038, A334301, A334302, A334439, A334442, A335122, A344085, A344086, A344088, A344089.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

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

cp's OEIS Frontend

A241918 Table of partitions where the ordering is based on the modified partial sums of the exponents of primes in the prime factorization of n.

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A334440 Irregular triangle T(n,k) read by rows: row n lists numbers of distinct parts of the n-th integer partition in Abramowitz-Stegun (sum/length/lex) order.

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

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A115729 Number of subpartitions of partitions in Mathematica order.

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A124832 Table of exponents of prime factorizations in A025487.

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