A036043 -id:A036043 - cp's OEIS frontend

A048996 Irregular triangle read by rows. Preferred multisets: numbers refining A007318 using format described in A036038.

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 2, 2, 3, 3, 4, 1, 1, 2, 2, 1, 3, 6, 1, 4, 6, 5, 1, 1, 2, 2, 2, 3, 6, 3, 3, 4, 12, 4, 5, 10, 6, 1, 1, 2, 2, 2, 1, 3, 6, 6, 3, 3, 4, 12, 6, 12, 1, 5, 20, 10, 6, 15, 7, 1, 1, 2, 2, 2, 2, 3, 6, 6, 3, 3, 6, 1, 4, 12, 12, 12, 12, 4, 5, 20, 10, 30, 5, 6, 30, 20, 7, 21, 8, 1

Offset: 0

Alford Arnold

This array gives in row n>=1 the multinomial numbers (call them M_0 numbers) m!/product((a_j)!,j=1..n) with the exponents of the partitions of n with number of parts m:=sum(a_j,j=1..n), given in the Abramowitz-Stegun order. See p. 831 of the given reference. See also the arrays for the M_1, M_2 and M_3 multinomial numbers A036038, A036039 and A036040 (or A080575).
For a signed version see A111786.
These M_0 multinomial numbers give the number of compositions of n >= 1 with parts corresponding to the partitions of n (in A-St order). See an n = 5 example below. The triangle with the summed entries of like number of parts m is A007318(n-1, m-1) (Pascal). - Wolfdieter Lang, Jan 29 2021

			Table starts:
[1]
[1]
[1, 1]
[1, 2, 1]
[1, 2, 1, 3, 1]
[1, 2, 2, 3, 3, 4, 1]
[1, 2, 2, 1, 3, 6, 1, 4, 6,  5, 1]
[1, 2, 2, 2, 3, 6, 3, 3, 4, 12, 4, 5, 10, 6, 1]
.
T(5,6) = 4 because there are four multisets using the first four digits {0,1,2,3}: 32100, 32110, 32210 and 33210
T(5,6) = 4 because there are 4 compositions of 5 that can be formed from the partition 2+1+1+1. - _Geoffrey Critzer_, May 19 2013
These 4 compositions 2+1+1+1, 1+2+1+1, 1+1+2+1 and 1+1+1+2 of 5 correspond to the 4 set partitions of [5] :={1,2,3,4,5}, with 4 blocks of consecutive numbers, namely {1,2},{3},{4},{5} and {1},{2,3},{4},{5} and {1},{2},{3,4},{5} and {1},{2},{3},{4,5}. - _Wolfdieter Lang_, May 30 2018

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1972.
Wolfdieter Lang, First 10 rows and more.

Cf. A000670, A007318, A036035, A036038, A019538, A115621, A309004, A000079 (row sums), A000041 (row lengths).

nmax:=9: with(combinat): for n from 1 to nmax do P(n):=sort(partition(n)): for r from 1 to numbpart(n) do B(r):=P(n)[r] od: for m from 1 to numbpart(n) do s:=0: j:=0: while sA036040(n, m) := (add(q(t), t=1..n))!/(mul(q(t)!, t=1..n)); od: od: seq(seq(A036040(n, m), m=1..numbpart(n)), n=1..nmax); # Johannes W. Meijer, Jul 14 2016

C(sig)={my(S=Set(sig)); (#sig)!/prod(k=1, #S, (#select(t->t==S[k], sig))!)}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 18 2020

from collections import Counter
def ASPartitions(n, k):
    Q = [p.to_list() for p in Partitions(n, length=k)]
    for q in Q: q.reverse()
    return sorted(Q)
def A048996_row(n):
    h = lambda p: product(map(factorial, Counter(p).values()))
    return [factorial(len(p))//h(p) for k in (0..n) for p in ASPartitions(n, k)]
for n in (1..10): print(A048996_row(n)) # Peter Luschny, Nov 02 2019 [corrected on notice from Sean A. Irvine, Apr 30 2022]

T(n,k) = A036040(n,k) * Factorial(A036043(n,k)) / A036038(n,k) = A049019(n,k) / A036038(n,k).
If the n-th partition is P, a(n) is the multinomial coefficient of the signature of P. - Franklin T. Adams-Watters, May 30 2006
T(n,k) = A309004(A036035(n,k)). - Andrew Howroyd, Oct 19 2020

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001

a(0)=1 prepended by Andrew Howroyd, Oct 19 2020

A334433 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally lexicographically.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 30 2020

First differs from A334435 at a(75) = 99, A334435(75) = 98.
A permutation of the positive integers.
This is the Abramowitz-Stegun ordering of integer partitions when the parts are read in the usual (weakly decreasing) order. The case of reversed (weakly increasing) partitions is A185974.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
As a triangle with row lengths A000041, the sequence starts {{1},{2},{3,4},{5,6,8},...}, so offset is 0.

			The sequence of terms together with their prime indices begins:
    1: {}            32: {1,1,1,1,1}       42: {1,2,4}
    2: {1}           13: {6}               44: {1,1,5}
    3: {2}           25: {3,3}             54: {1,2,2,2}
    4: {1,1}         21: {2,4}             60: {1,1,2,3}
    5: {3}           22: {1,5}             56: {1,1,1,4}
    6: {1,2}         27: {2,2,2}           72: {1,1,1,2,2}
    8: {1,1,1}       30: {1,2,3}           80: {1,1,1,1,3}
    7: {4}           28: {1,1,4}           96: {1,1,1,1,1,2}
    9: {2,2}         36: {1,1,2,2}        128: {1,1,1,1,1,1,1}
   10: {1,3}         40: {1,1,1,3}         19: {8}
   12: {1,1,2}       48: {1,1,1,1,2}       49: {4,4}
   16: {1,1,1,1}     64: {1,1,1,1,1,1}     55: {3,5}
   11: {5}           17: {7}               39: {2,6}
   15: {2,3}         35: {3,4}             34: {1,7}
   14: {1,4}         33: {2,5}             75: {2,3,3}
   18: {1,2,2}       26: {1,6}             63: {2,2,4}
   20: {1,1,3}       45: {2,2,3}           70: {1,3,4}
   24: {1,1,1,2}     50: {1,3,3}           66: {1,2,5}
Triangle begins:
   1
   2
   3   4
   5   6   8
   7   9  10  12  16
  11  15  14  18  20  24  32
  13  25  21  22  27  30  28  36  40  48  64
  17  35  33  26  45  50  42  44  54  60  56  72  80  96 128
This corresponds to the tetrangle:
                  0
                 (1)
               (2)(11)
             (3)(21)(111)
        (4)(22)(31)(211)(1111)
  (5)(32)(41)(221)(311)(2111)(11111)

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

Row lengths are A000041.
Compositions under the same order are A124734 (triangle).
The version for reversed (weakly increasing) partitions is A185974.
The constructive version is A334301.
Ignoring length gives A334434, or A334437 for reversed partitions.
The dual version (sum/length/revlex) is A334438.
Lexicographically ordered reversed partitions are A026791.
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.
Graded Heinz numbers are A215366.
Sorting partitions by Heinz number gives A296150.
Partitions in increasing-length reverse-lexicographic order (sum/length/revlex) are A334439 (not A036037).
Cf. A026791, A036043, A048793, A056239, A129129, A211992, A228351, A228531, A334302, A334435, A334436.

Join@@Table[Times@@Prime/@#&/@Sort[IntegerPartitions[n]],{n,0,8}]

A001222(a(n)) = A036043(n).

A334302 Irregular triangle read by rows where row k is the k-th reversed integer partition, if reversed partitions are sorted first by sum, then by length, and finally reverse-lexicographically.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 30 2020

			The sequence of all reversed partitions begins:
  ()         (1,4)        (1,1,1,1,2)
  (1)        (1,2,2)      (1,1,1,1,1,1)
  (2)        (1,1,3)      (7)
  (1,1)      (1,1,1,2)    (3,4)
  (3)        (1,1,1,1,1)  (2,5)
  (1,2)      (6)          (1,6)
  (1,1,1)    (3,3)        (2,2,3)
  (4)        (2,4)        (1,3,3)
  (2,2)      (1,5)        (1,2,4)
  (1,3)      (2,2,2)      (1,1,5)
  (1,1,2)    (1,2,3)      (1,2,2,2)
  (1,1,1,1)  (1,1,4)      (1,1,2,3)
  (5)        (1,1,2,2)    (1,1,1,4)
  (2,3)      (1,1,1,3)    (1,1,1,2,2)
This sequence can also be interpreted as the following triangle, whose n-th row is itself a finite triangle with A000041(n) rows.
                            0
                           (1)
                        (2) (1,1)
                    (3) (1,2) (1,1,1)
            (4) (2,2) (1,3) (1,1,2) (1,1,1,1)
  (5) (2,3) (1,4) (1,2,2) (1,1,3) (1,1,1,2) (1,1,1,1,1)
Showing partitions as their Heinz numbers (see A334435) gives:
   1
   2
   3   4
   5   6   8
   7   9  10  12  16
  11  15  14  18  20  24  32
  13  25  21  22  27  30  28  36  40  48  64
  17  35  33  26  45  50  42  44  54  60  56  72  80  96 128

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

Row lengths are A036043.
Lexicographically ordered reversed partitions are A026791.
The dual ordering (sum/length/lex) of reversed partitions is A036036.
Reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Lexicographically ordered partitions are A193073.
Graded Heinz numbers are A215366.
Ignoring length gives A228531.
Sorting partitions by Heinz number gives A296150.
The version for compositions is A296774.
The dual ordering (sum/length/lex) of non-reversed partitions is A334301.
Taking Heinz numbers gives A334435.
The version for regular (non-reversed) partitions is A334439 (not A036037).
Cf. A000041, A048793, A066099, A080576, A124734, A162247, A211992, A228100, A228351.

revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]

A334435 Heinz numbers of all reversed integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

Original entry on oeis.org

Offset: 0

Gus Wiseman, May 02 2020

First differs from A334433 at a(75) = 99, A334433(75) = 98.
First differs from A334436 at a(22) = 22, A334436(22) = 27.
A permutation of the positive integers.
Reversed integer partitions are finite weakly increasing sequences of positive integers.
This is the Abramowitz-Stegun ordering of reversed partitions (A185974) except that the finer order is reverse-lexicographic instead of lexicographic. The version for non-reversed partitions is A334438.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
As a triangle with row lengths A000041, the sequence starts {{1},{2},{3,4},{5,6,8},...}, so offset is 0.

			The sequence of terms together with their prime indices begins:
    1: {}            32: {1,1,1,1,1}       42: {1,2,4}
    2: {1}           13: {6}               44: {1,1,5}
    3: {2}           25: {3,3}             54: {1,2,2,2}
    4: {1,1}         21: {2,4}             60: {1,1,2,3}
    5: {3}           22: {1,5}             56: {1,1,1,4}
    6: {1,2}         27: {2,2,2}           72: {1,1,1,2,2}
    8: {1,1,1}       30: {1,2,3}           80: {1,1,1,1,3}
    7: {4}           28: {1,1,4}           96: {1,1,1,1,1,2}
    9: {2,2}         36: {1,1,2,2}        128: {1,1,1,1,1,1,1}
   10: {1,3}         40: {1,1,1,3}         19: {8}
   12: {1,1,2}       48: {1,1,1,1,2}       49: {4,4}
   16: {1,1,1,1}     64: {1,1,1,1,1,1}     55: {3,5}
   11: {5}           17: {7}               39: {2,6}
   15: {2,3}         35: {3,4}             34: {1,7}
   14: {1,4}         33: {2,5}             75: {2,3,3}
   18: {1,2,2}       26: {1,6}             63: {2,2,4}
   20: {1,1,3}       45: {2,2,3}           70: {1,3,4}
   24: {1,1,1,2}     50: {1,3,3}           66: {1,2,5}
Triangle begins:
   1
   2
   3   4
   5   6   8
   7   9  10  12  16
  11  15  14  18  20  24  32
  13  25  21  22  27  30  28  36  40  48  64
  17  35  33  26  45  50  42  44  54  60  56  72  80  96 128
This corresponds to the following tetrangle:
                  0
                 (1)
               (2)(11)
             (3)(12)(111)
        (4)(22)(13)(112)(1111)
  (5)(23)(14)(122)(113)(1112)(11111)

Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
The dual version (sum/length/lex) is A185974.
Compositions under the same order are A296774 (triangle).
The constructive version is A334302.
Ignoring length gives A334436.
The version for non-reversed partitions is A334438.
Partitions in this order (sum/length/revlex) are A334439.
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.
Reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Graded lexicographically ordered partitions are A193073.
Partitions in colexicographic (sum/colex) order are A211992.
Graded Heinz numbers are given by A215366.
Sorting partitions by Heinz number gives A296150.
Cf. A056239, A124734, A129129, A228100, A228531, A333219, A333220, A334301, A334433, A334434, A334437.

revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]

A001222(a(n)) = A036043(n).

A228100 Triangle in which n-th row lists all partitions of n, such that partitions of n into m parts appear in lexicographic order previous to the partitions of n into k parts if k < m. (Fenner-Loizou tree.)

Original entry on oeis.org

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

Offset: 1

Peter Luschny, Aug 10 2013

First differs from A193073 at a(58). - Omar E. Pol, Sep 22 2013

The partition lengths appear to be A331581. - Gus Wiseman, May 12 2020

			The sixth row is:
[1, 1, 1, 1, 1, 1]
[2, 1, 1, 1, 1]
[2, 2, 1, 1]
[3, 1, 1, 1]
[2, 2, 2]
[3, 2, 1]
[4, 1, 1]
[3, 3]
[4, 2]
[5, 1]
[6]
From _Gus Wiseman_, May 10 2020: (Start)
The triangle with partitions shown as Heinz numbers (A333485) begins:
    1
    2
    4   3
    8   6   5
   16  12   9  10   7
   32  24  18  20  15  14  11
   64  48  36  40  27  30  28  25  21  22  13
  128  96  72  80  54  60  56  45  50  42  44  35  33  26  17
(End)

T. I. Fenner, G. Loizou: A binary tree representation and related algorithms for generating integer partitions. The Computer J. 23(4), 332-337 (1980)
D. E. Knuth: The Art of Computer Programming. Generating all combinations and partitions, vol. 4, fasc. 3, 7.2.1.4, exercise 10.
K. Yamanaka, Y. Otachi, Sh. Nakano: Efficient enumeration of ordered trees with k leaves. In: WALCOM: Algorithms and Computation, Lecture Notes in Computer Science Volume 5431, 141-150 (2009)
S. Zaks, D. Richards: Generating trees and other combinatorial objects lexicographically. SIAM J. Comput. 8(1), 73-81 (1979)
A. Zoghbi, I. Stojmenovic': Fast algorithms for generating integer partitions. Int. J. Comput. Math. 70, 319-332 (1998)

Alois P. Heinz, rows n = 1..19, flattened
Peter Luschny, Integer Partition Trees
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

See A036036 for the Hindenburg (graded reflected colexicographic) ordering.
See A036037 for the graded colexicographic ordering.
See A080576 for the Maple (graded reflected lexicographic) ordering.
See A080577 for the Mathematica (graded reverse lexicographic) ordering.
See A182937 the Fenner-Loizou (binary tree in preorder traversal) ordering.
See A193073 for the graded lexicographic ordering.
The version for compositions is A296773.
Taking Heinz numbers gives A333485.
Lexicographically ordered reversed partitions are A026791.
Sorting partitions by Heinz number gives A296150, or A112798 for reversed partitions.
Reversed partitions under the (sum/length/revlex) ordering are A334302.
Cf. A000041, A036043, A048793, A211992, A228351, A228531, A331581, A333484, A334301, A334439.

b:= proc(n, i) b(n, i):= `if`(n=0 or i=1, [[1$n]], [b(n, i-1)[],
      `if`(i>n, [], map(x-> [i, x[]], b(n-i, i)))[]])
    end:
T:= n-> map(h-> h[], sort(b(n$2), proc(x, y) local i;
        if nops(x)<>nops(y) then return nops(x)>nops(y) else
        for i to nops(x) do if x[i]<>y[i] then return x[i]Alois P. Heinz, Aug 13 2013

row[n_] := Flatten[Reverse[Sort[#]]& /@ SplitBy[Sort[IntegerPartitions[n] ], Length], 1] // Reverse; Array[row, 8] // Flatten (* Jean-François Alcover, Dec 05 2016 *)
ralensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]>Length[c],OrderedQ[{f,c}]];
Join@@Table[Sort[IntegerPartitions[n],ralensort],{n,0,8}] (* Gus Wiseman, May 10 2020 *)

from collections import deque
def GeneratePartitions(n, visit):
    p = ([], 0, n)
    queue = deque()
    queue.append(p)
    visit(p)
    while len(queue) > 0 :
        (phead, pheadLen, pnum1s) = queue.popleft()
        if pnum1s != 1 :
            head = phead[:pheadLen] + [2]
            q = (head, pheadLen + 1, pnum1s - 2)
            if 1 <= q[2] : queue.append(q)
            visit(q)
        if pheadLen == 1 or (pheadLen > 1 and \
                      (phead[pheadLen - 1] != phead[pheadLen - 2])) :
            head = phead[:pheadLen]
            head[pheadLen - 1] += 1
            q = (head, pheadLen, pnum1s - 1)
            if 1 <= q[2] : queue.append(q)
            visit(q)
def visit(q): print(q[0] + [1 for i in range(q[2])])
for n in (1..7): GeneratePartitions(n, visit)

A334438 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 03 2020

First differs from A185974 shifted left once at a(76) = 99, A185974(75) = 98.
A permutation of the positive integers.
This is the Abramowitz-Stegun ordering of integer partitions (A334433) except that the finer order is reverse-lexicographic instead of lexicographic. The version for reversed partitions is A334435.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
As a triangle with row lengths A000041, the sequence starts {{1},{2},{3,4},{5,6,8},...}, so offset is 0.

			The sequence of terms together with their prime indices begins:
    1: {}            32: {1,1,1,1,1}       50: {1,3,3}
    2: {1}           13: {6}               45: {2,2,3}
    3: {2}           22: {1,5}             56: {1,1,1,4}
    4: {1,1}         21: {2,4}             60: {1,1,2,3}
    5: {3}           25: {3,3}             54: {1,2,2,2}
    6: {1,2}         28: {1,1,4}           80: {1,1,1,1,3}
    8: {1,1,1}       30: {1,2,3}           72: {1,1,1,2,2}
    7: {4}           27: {2,2,2}           96: {1,1,1,1,1,2}
   10: {1,3}         40: {1,1,1,3}        128: {1,1,1,1,1,1,1}
    9: {2,2}         36: {1,1,2,2}         19: {8}
   12: {1,1,2}       48: {1,1,1,1,2}       34: {1,7}
   16: {1,1,1,1}     64: {1,1,1,1,1,1}     39: {2,6}
   11: {5}           17: {7}               55: {3,5}
   14: {1,4}         26: {1,6}             49: {4,4}
   15: {2,3}         33: {2,5}             52: {1,1,6}
   20: {1,1,3}       35: {3,4}             66: {1,2,5}
   18: {1,2,2}       44: {1,1,5}           70: {1,3,4}
   24: {1,1,1,2}     42: {1,2,4}           63: {2,2,4}
Triangle begins:
   1
   2
   3   4
   5   6   8
   7  10   9  12  16
  11  14  15  20  18  24  32
  13  22  21  25  28  30  27  40  36  48  64
  17  26  33  35  44  42  50  45  56  60  54  80  72  96 128
This corresponds to the following tetrangle:
                  0
                 (1)
               (2)(11)
             (3)(21)(111)
        (4)(31)(22)(211)(1111)
  (5)(41)(32)(311)(221)(2111)(11111)

Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
Ignoring length gives A129129.
Compositions under the same order are A296774 (triangle).
The dual version (sum/length/lex) is A334433.
The version for reversed partitions is A334435.
The constructive version is A334439 (triangle).
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colexicographic order (sum/length/colex) are A036037.
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.
Cf. A056239, A066099, A124734, A185974, A228100, A228531, A333219, A334301, A334302, A334434, A334436, A334437.

revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]

A001221(a(n)) = A103921(n).

A001222(a(n)) = A036043(n).

A103921 Irregular triangle T(n,m) (n >= 0) read by rows: row n lists numbers of distinct parts of partitions of n in Abramowitz-Stegun order.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Mar 24 2005

T(n, m) is the number of distinct parts of the m-th partition of n in Abramowitz-Stegun order; n >= 0, m = 1..p(n) = A000041(n).
The row length sequence of this table is p(n)=A000041(n) (number of partitions).
In order to count distinct parts of a partition consider the partition as a set instead of a multiset. E.g., n=6: read [1,1,1,3] as {1,3} and count the elements, here 2.
Rows are the same as the rows of A115623, but in reverse order.
From Wolfdieter Lang, Mar 17 2011: (Start)
The number of 1s in row number n, n >= 1, is tau(n)=A000005(n), the number of divisors of n.
For the proof read off the divisors d(n,j), j=1..tau(n), from row number n of table A027750, and translate them to the tau(n) partitions d(n,1)^(n/d(n,1)), d(n,2)^(n/d(n,2)),..., d(n,tau(n))^(n/d(n,tau(n))).
See a comment by Giovanni Resta under A000005. (End)
From Gus Wiseman, May 20 2020: (Start)
The name is correct if integer partitions are read in reverse, so that the parts are weakly increasing. The non-reversed version is A334440.
Also the number of distinct parts of the n-th integer partition in lexicographic order (A193073).
Differs from the number of distinct parts in the n-th integer partition in (sum/length/revlex) order (A334439). For example, (6,2,2) has two distinct elements, while (1,4,5) has three.
(End)

			Triangle starts:
  0,
  1,
  1, 1,
  1, 2, 1,
  1, 2, 1, 2, 1,
  1, 2, 2, 2, 2, 2, 1,
  1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1,
  1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 1,
  1, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 3, 2, 3, 1, 2, 3, 2, 2, 2, 2, 1,
  1, 2, 2, 2, 2, ...
a(5,4)=2 from the fourth partition of 5 in the mentioned order, i.e., (1^2,3), which has two distinct parts, namely 1 and 3.

Robert Price, Table of n, a(n) for n = 0..9295 (first 25 rows).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
Wolfdieter Lang, First 10 rows.
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row sums are A000070.
Row lengths are A000041.
The lengths of these partitions are A036043.
The maxima of these partitions are A049085.
The version for non-reversed partitions is A334440.
The version for colex instead of lex is (also) A334440.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun order are A036036.
Reverse-lexicographically ordered partitions are A080577.
Compositions in Abramowitz-Stegun order are A124734.
Cf. A001221, A036037, A112798, A115621, A115623, A185974, A193073, A228531, A334301, A334302, A334433, A334435, A334441.

Join@@Table[Length/@Union/@Sort[Reverse/@IntegerPartitions[n]],{n,0,8}] (* Gus Wiseman, May 20 2020 *)

a(n) = A001221(A185974(n)). - Gus Wiseman, May 20 2020

Edited by Franklin T. Adams-Watters, May 29 2006

A334442 Irregular triangle whose reversed rows are all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 07 2020

First differs from A036036 for reversed partitions of 9. Namely, this sequence has (2,2,5) before (1,4,4), while A036036 has (1,4,4) before (2,2,5).

			The sequence of all partitions begins:
  ()         (2,3)        (1,1,1,1,2)    (1,1,1,2,2)
  (1)        (1,1,3)      (1,1,1,1,1,1)  (1,1,1,1,1,2)
  (2)        (1,2,2)      (7)            (1,1,1,1,1,1,1)
  (1,1)      (1,1,1,2)    (1,6)          (8)
  (3)        (1,1,1,1,1)  (2,5)          (1,7)
  (1,2)      (6)          (3,4)          (2,6)
  (1,1,1)    (1,5)        (1,1,5)        (3,5)
  (4)        (2,4)        (1,2,4)        (4,4)
  (1,3)      (3,3)        (1,3,3)        (1,1,6)
  (2,2)      (1,1,4)      (2,2,3)        (1,2,5)
  (1,1,2)    (1,2,3)      (1,1,1,4)      (1,3,4)
  (1,1,1,1)  (2,2,2)      (1,1,2,3)      (2,2,4)
  (5)        (1,1,1,3)    (1,2,2,2)      (2,3,3)
  (1,4)      (1,1,2,2)    (1,1,1,1,3)    (1,1,1,5)
This sequence can also be interpreted as the following triangle:
                  0
                 (1)
               (2)(11)
             (3)(12)(111)
        (4)(13)(22)(112)(1111)
  (5)(14)(23)(113)(122)(1112)(11111)
Taking Heinz numbers (A334438) gives:
   1
   2
   3   4
   5   6   8
   7  10   9  12  16
  11  14  15  20  18  24  32
  13  22  21  25  28  30  27  40  36  48  64
  17  26  33  35  44  42  50  45  56  60  54  80  72  96 128

Wikiversity, Lexicographic and colexicographic order

Row lengths are A036043.
The version for reversed partitions is A334301.
The version for colex instead of revlex is A334302.
Taking Heinz numbers gives A334438.
The version with rows reversed is A334439.
Ignoring length gives A335122.
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.
Reverse-lexicographically ordered partitions are A080577.
Lexicographically ordered partitions are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Sorting partitions by Heinz number gives A296150.
Cf. A026791, A112798, A124734, A129129, A185974, A228100, A228531, A296774, A334433, A334435, A334436.

revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]

A334442_row(n)=vecsort(partitions(n),p->concat(#p,-Vecrev(p))) \\ Rows of triangle defined in EXAMPLE (all partitions of n). Wrap into [Vec(p)|p<-...] to avoid "Vecsmall". - M. F. Hasler, May 14 2020

A334441 Maximum part of the n-th integer partition in Abramowitz-Stegun (sum/length/lex) order; a(0) = 0.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 06 2020

First differs from A049085 at a(8) = 2, A049085(8) = 3.

The parts of a partition are read in the usual (weakly decreasing) order. The version for reversed (weakly increasing) partitions is A049085.

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

Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
The length of the same partition is A036043.
Ignoring partition length (sum/lex) gives A036043 also.
The version for reversed partitions is A049085.
a(n) is the maximum element in row n of A334301.
The number of distinct parts in the same partition is A334440.
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, A103921, A124734, A185974, A296774, A299755, A334302, A334433, A334434, A334435, A334438.

Table[If[n==0,{0},Max/@Sort[IntegerPartitions[n]]],{n,0,10}]

A049019 Irregular triangle read by rows: Row n gives numbers of preferential arrangements (onto functions) of n objects that are associated with the partition of n, taken in Abramowitz and Stegun order.

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 1, 8, 6, 36, 24, 1, 10, 20, 60, 90, 240, 120, 1, 12, 30, 20, 90, 360, 90, 480, 1080, 1800, 720, 1, 14, 42, 70, 126, 630, 420, 630, 840, 5040, 2520, 4200, 12600, 15120, 5040, 1, 16, 56, 112, 70, 168, 1008, 1680, 1260, 1680, 1344, 10080, 6720

Offset: 1

Alford Arnold

This is a refinement of A019538 with row sums in A000670.
From Tom Copeland, Sep 29 2008: (Start)
This array is related to the reciprocal of an e.g.f. as sketched in A133314. For example, the coefficient of the fourth-order term in the Taylor series expansion of 1/(a(0) + a(1) x + a(2) x^2/2! + a(3) x^3/3! + ...) is a(0)^(-5) * {24 a(1)^4 - 36 a(1)^2 a(2) a(0) + [8 a(1) a(3) + 6 a(2)^2] a(0)^2 - a(4) a(0)^3}.
The unsigned coefficients characterize the P3 permutohedron depicted on page 10 in the Loday link with 24 vertices (0-D faces), 36 edges (1-D faces), 6 squares (2-D faces), 8 hexagons (2-D faces) and 1 3-D permutohedron. Summing coefficients over like dimensions gives A019538 and A090582. Compare to A133437 for the associahedron.
Given the n X n lower triangular matrix M = [ binomial(j,k) u(j-k) ], the first column of the inverse matrix M^(-1) contains the (n-1) rows of A049019 as the coefficients of the multinomials formed from the u(j). M^(-1) can be computed as (1/u(0)){I - [I- M/u(0)]^n} / {I - [I- M/u(0)]} = - u(0)^(-n) {sum(j=1 to n)(-1)^j bin(n,j) u(0)^(n-j) M^(j-1)} where I is the identity matrix.
Another method for computing the coefficients and partitions up to (n-1) rows is to use (1-x^n)/ (1-x) = 1+x^2+x^3+ ... + x^(n-1) with x replaced either by [I- M/a(0)] or [1- g(x)/a(0)] with the n X n matrix M = [bin(j,k) a(j-k)] and g(x)= a(0) + a(1)x + a(2)x^2/2! + ... + a(n) x^n/n!. The first n terms (rows of the first column) of the resulting series (matrix) divided by a(0) contain the (n-1) rows of signed coefficients and associated partitions for A049019.
To obtain unsigned coefficients, change a(j) to -a(j) for j>0. A133314 contains other matrices and recursion formulas that could be used. The Faa di Bruno formula gives the coefficients as n! [e(1)+e(2)+...+e(n)]! / [1!^e(1) e(1)! 2!^e(2) e(2)!... n!^e(n) e(n)! ] for the partition of form [a(1)^e(1)...a(n)^e(n)] with [e(1)+2e(2)+...+ n e(n)] = n (see Abramowitz and Stegun pages 823 and 831) in agreement with Arnold's formula. (End)

			Irregular triangle starts (note the grouping by ';' when comparing with A019538):
[1] 1;
[2] 1;  2;
[3] 1;  6;  6;
[4] 1;  8,  6; 36;  24;
[5] 1; 10, 20; 60,  90; 240; 120;
[6] 1; 12, 30, 20;  90, 360,  90; 480, 1080; 1800; 720;
[7] 1; 14, 42, 70; 126, 630, 420, 630;  840, 5040, 2520; 4200, 12600; 15120; 5040;
.
a(17) = 240 because we can write
A048996(17)*A036038(17) = 4*60 = A036040(17)*A036043(17)! = 10*24.
As in A133314, 1/exp[u(.)*x] = u(0)^(-1) [ 1 ] + u(0)^(-2) [ -u(1) ] x + u(0)^(-3) [ -u(0)u(2) + 2 u(1)^2 ] x^2/2! + u(0)^(-4) [ -u(0)^2 u(3) + 6 u(0)u(1)u(2) - 6 u(1)^3 ] x^3/3! + u(0)^(-5) [ -u(0)^3 u(4) + 8 u(0)^2 u(1)u(3) + 6 u(0)^2 u(2)^2 - 36 u(0)u(1)^2 u(2) + 24 u(1)^4 ] x^4/4! + ... . These are essentially refined face polynomials for permutohedra: empty set + point + line segment + hexagon + 3-D- permutohedron + ... . - _Tom Copeland_, Oct 04 2008

Peter Luschny, Rows n = 1..36, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
N. Arkani-Hamed, Y. Bai, S. He, and G. Yan, Scattering forms and the positive geometry of kinematics, color, and the worldsheet , arXiv:1711.09102 [hep-th], 2017.
Tom Copeland, Bijective mapping between face polytopes of permutohedra and partitions of integers, Math StackExchange question, 2016.
S. Forcey, The Hedra Zoo.
X. Gao, S. He, and Y. Zhan, Labelled tree graphs, Feynman diagrams and disk integrals , arXiv:1708.08701 [hep-th], 2017.
J. Loday, The Multiple Facets of the Associahedron
V. Pilaud, The Associahedron and its Friends, presentation for Seminaire Lotharingien de Combinatoire, April 4 - 6, 2016.
A. Postnikov, Positive Grassmannian and Polyhedral Subdivisions, arXiv:1806.05307 [math.CO], (cf. p. 17), 2018.

Cf. A000041, A000670, A019538, A036038, A036040, A036043, A048996, A133314.

def A049019(n):
    if n == 0: return [1]
    P = lambda k: Partitions(n, min_length=k, max_length=k)
    Q = (p.to_list() for k in (1..n) for p in P(k))
    return [factorial(len(p))*SetPartitions(sum(p), p).cardinality() for p in Q]
for n in (1..7): print(A049019(n)) # Peter Luschny, Aug 30 2019

a(n) = A048996(n) * A036038(n);
a(n) = A036040(n) * factorial(A036043(n)).
A lowering operator for the unsigned multinomials in the brackets in the example is [d/du(1) 1/POP] where u(1) is treated as a continuous variable and POP is an operator that pulls off the number of parts of a partition ignoring u(0), e.g., [d/du(1) 1/POP][ u(0)u(2) + 2 u(1)^2 ] = (1/2) 2*2 u(1) = 2*u(1), analogous to the prototypical delta operator (d/dz) z^n = n z^(n-1). - Tom Copeland, Oct 04 2008
From the matrix formulation with M_m,k = 1/(m-k)!; g(x) = exp[ u(.) x]; an orthonormal vector basis x_1, ..., x_n and En(x^k) = x_k for k <= n and zero otherwise, for j=0 to n-1 the j-th signed row multinomial is given by the wedge product of x_1 with the wedge product (-1)^j * j! * u(0)^(-n) * Wedge{ En[x g(x), x^2 g(x), ..., x^(j) g(x), ~, x^(j+2) g(x), ..., x^n g(x)] } where Wedge{a,b,c} = a v b v c (the usual wedge symbol is inverted here to prevent confusion with the power notation, see Mathworld) and the (j+1)-th element is omitted from the product. Tom Copeland, Oct 06 2008 [Changed an x^n to x^(n-1) and "inner product of x_1" to "wedge". - Tom Copeland, Feb 03 2010]

Partitions for 7 and 8 from Tom Copeland, Oct 02 2008

Definition edited by N. J. A. Sloane, Nov 06 2023

cp's OEIS Frontend

A048996 Irregular triangle read by rows. Preferred multisets: numbers refining A007318 using format described in A036038.

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A334433 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally lexicographically.

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A334302 Irregular triangle read by rows where row k is the k-th reversed integer partition, if reversed partitions are sorted first by sum, then by length, and finally reverse-lexicographically.

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A334435 Heinz numbers of all reversed integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

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A228100 Triangle in which n-th row lists all partitions of n, such that partitions of n into m parts appear in lexicographic order previous to the partitions of n into k parts if k < m. (Fenner-Loizou tree.)

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A334438 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

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A103921 Irregular triangle T(n,m) (n >= 0) read by rows: row n lists numbers of distinct parts of partitions of n in Abramowitz-Stegun order.

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