A036036 -id:A036036 - cp's OEIS frontend

A036038 Triangle of multinomial coefficients.

1, 1, 2, 1, 3, 6, 1, 4, 6, 12, 24, 1, 5, 10, 20, 30, 60, 120, 1, 6, 15, 20, 30, 60, 90, 120, 180, 360, 720, 1, 7, 21, 35, 42, 105, 140, 210, 210, 420, 630, 840, 1260, 2520, 5040, 1, 8, 28, 56, 70, 56, 168, 280, 420, 560, 336, 840, 1120, 1680, 2520, 1680, 3360, 5040, 6720

Offset: 1

N. J. A. Sloane

The number of terms in the n-th row is the number of partitions of n, A000041(n). - Amarnath Murthy, Sep 21 2002
For each n, the partitions are ordered according to A-St: first by length and then lexicographically (arranging the parts in nondecreasing order), which is different from the usual practice of ordering all partitions lexicographically. - T. D. Noe, Nov 03 2006
For this ordering of the partitions, for n >= 1, see the remarks and the C. F. Hindenburg link given in A036036. - Wolfdieter Lang, Jun 15 2012
The relation (n+1) * A134264(n+1) = A248120(n+1) / a(n) where the arithmetic is performed for matching partitions in each row n connects the combinatorial interpretations of this array to some topological and algebraic constructs of the two other entries. Also, these seem (cf. MOPS reference, Table 2) to be the coefficients of the Jack polynomial J(x;k,alpha=0). - Tom Copeland, Nov 24 2014
The conjecture on the Jack polynomials of zero order is true as evident from equation a) on p. 80 of the Stanley reference, suggested to me by Steve Kass. The conventions for denoting the more general Jack polynomials J(n,alpha) vary. Using Stanley's convention, these Jack polynomials are the umbral extensions of the multinomial expansion of (s_1*x_1 + s_2*x_2 + ... + s_(n+1)*x_(n+1))^n in which the subscripts of the (s_k)^j in the symmetric monomial expansions are finally ignored and the exponent dropped to give s_j(alpha) = j-th row polynomial of A094638 or |A008276| in ascending powers of alpha. (The MOPS table has some inconsistency between n = 3 and n = 4.) - Tom Copeland, Nov 26 2016

			1;
1, 2;
1, 3,  6;
1, 4,  6, 12, 24;
1, 5, 10, 20, 30, 60, 120;
1, 6, 15, 20, 30, 60,  90, 120, 180, 360, 720;

Abramowitz and Stegun, Handbook, p. 831, column labeled "M_1".

David W. Wilson, Table of n, a(n) for n = 1..11731 (rows 1 through 26).
Milton Abramowitz and Irene A. Stegun, editors, Multinomials: M_1, M_2 and M_3, Handbook of Mathematical Functions, December 1972, pp. 831-2.
T. Copeland, Witt Differential Generator for Special Jack Symmetric Functions / Polynomials, 2016.
I. Dumitriu, A. Edelman, G. Schuman, MOPS: Multivariate orthogonal polynomials (symbolically), arxiv:0409066 [math-ph], 2004.
Wolfdieter Lang, First 10 rows and more.
R. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. in Math., 77, p. 76-115, 1989.

Cf. A036036-A036040. Different from A078760. Row sums give A005651.
Cf. A183610 is a table of sums of powers of terms in rows.
Cf. A134264 and A248120.
Cf. A008275, A008276, A094638, A248927.
Cf. A096162 for connections to A130561.

nmax:=7: 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 sA036038(n, m) := n!/ (mul((t!)^q(t), t=1..n)); od: od: seq(seq(A036038(n, m), m=1..numbpart(n)), n=1..nmax); # Johannes W. Meijer, Jul 14 2016

Flatten[Table[Apply[Multinomial, Reverse[Sort[IntegerPartitions[i],  Length[ #1]>Length[ #2]&]], {1}], {i,9}]] (* T. D. Noe, Nov 03 2006 *)

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 A036038_row(n):
    return [multinomial(p) for k in (0..n) for p in ASPartitions(n, k)]
for n in (1..10): print(A036038_row(n))
# Peter Luschny, Dec 18 2016, corrected Apr 30 2022

The n-th row is the expansion of (x_1 + x_2 + ... + x_(n+1))^n in the basis of the monomial symmetric polynomials (m.s.p.). E.g., (x_1 + x_2 + x_3 + x_4)^3 = m[3](x_1,..,x_4) + 3*m[1,2](x_1,..,x_4) + 6*m[1,1,1](x_1,..,x_4) = (Sum_{i=1..4} x_i^3) + 3*(Sum_{i,j=1..4;i != j} x_i^2 x_j) + 6*(Sum_{i,j,k=1..4;i < j < k} x_i x_j x_k). The number of indeterminates can be increased indefinitely, extending each m.s.p., yet the expansion coefficients remain the same. In each m.s.p., unique combinations of exponents and subscripts appear only once with a coefficient of unity. Umbral reduction by replacing x_k^j with x_j in the expansions gives the partition polynomials of A248120. - Tom Copeland, Nov 25 2016
From Tom Copeland, Nov 26 2016: (Start)
As an example of the umbral connection to the Jack polynomials: J(3,alpha) = (Sum_{i=1..4} x_i^3)*s_3(alpha) + 3*(Sum_{i,j=1..4;i!=j} x_i^2 x_j)*s_2(alpha)*s_1(alpha)+ 6*(Sum_{i,j,k=1..4;i < j < k} x_i x_j x_k)*s_1(alpha)*s_1(alpha)*s_1(alpha) = (Sum_{i=1..4} x_i^3)*(1+alpha)*(1+2*alpha)+ 3*(sum_{i,j=1..4;i!=j} x_i^2 x_j)*(1+alpha) + 6*(Sum_{i,j,k=1..4;i < j < k} x_i x_j x_k).
See the Copeland link for more relations between the multinomial coefficients and the Jack symmetric functions. (End)

More terms from David W. Wilson and Wouter Meeussen

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

Original entry on oeis.org

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

A036039 Irregular triangle of multinomial coefficients of integer partitions read by rows (in Abramowitz and Stegun ordering) giving the coefficients of the cycle index polynomials for the symmetric groups S_n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 8, 3, 6, 1, 24, 30, 20, 20, 15, 10, 1, 120, 144, 90, 40, 90, 120, 15, 40, 45, 15, 1, 720, 840, 504, 420, 504, 630, 280, 210, 210, 420, 105, 70, 105, 21, 1, 5040, 5760, 3360, 2688, 1260, 3360, 4032, 3360, 1260, 1120, 1344, 2520, 1120, 1680, 105, 420, 1120, 420, 112, 210, 28, 1

Offset: 1

N. J. A. Sloane

The sequence of row lengths is A000041(n), n >= 1 (partition numbers).
Number of permutations whose cycle structure is the given partition. Row sums are factorials (A000142). - Franklin T. Adams-Watters, Jan 12 2006
A relation between partition polynomials formed from these "refined" Stirling numbers of the first kind and umbral operator trees and Lagrange inversion is presented in the link "Lagrange a la Lah".
These cycle index polynomials for the symmetric group S_n are also related to a raising operator / infinitesimal generator for fractional integro-derivatives, involving the digamma function and the Riemann zeta function values at positive integers, and to the characteristic polynomial for the adjacency matrix of complete n-graphs A055137 (cf. MathOverflow link). - Tom Copeland, Nov 03 2012
In the Lang link, replace all x(n) by t to obtain A132393. Furthermore replace x(1) by t and all other x(n) by 1 to obtain A008290. See A274760. - Tom Copeland, Nov 06 2012, Oct 29 2015 - corrected by Johannes W. Meijer, Jul 28 2016
The umbral compositional inverses of these polynomials are formed by negating the indeterminates x(n) for n>1, i.e., P(n,P(.,x(1),-x(2),-x(3),...),x(2),x(3),...) = x(1)^n (cf. A130561 for an example of umbral compositional inversion). The polynomials are an Appell sequence in x(1), i.e., dP(n,x(1))/dx(1) = n P(n-1, x(1)) and (P(.,x)+y)^n=P(n,x+y) umbrally, with P(0,x(1))=1. - Tom Copeland, Nov 14 2014
Regarded as the coefficients of the partition polynomials listed by Lang, a signed version of these polynomials IF(n,b1,b2,...,bn) (n! times polynomial on page 184 of Airault and Bouali) provides an inversion of the Faber polynomials F(n,b1,b2,...,bn) (page 52 of Bouali, A263916, and A115131). For example, F(3, IF(1,b1), IF(2,b1,b2)/2!, IF(3,b1,b2,b3)/3!) = b3 and IF(3, F(1,b1), F(2,b1,b2), F(3,b1,b2,b3))/3! = b3 with F(1,b1) = -b1. (Compare with A263634.) - Tom Copeland, Oct 28 2015; Sep 09 2016
The e.g.f. for the row partition polynomials is Sum_{n>=0} P_n(b_1,...,b_n) x^n/n! = exp[Sum_{n>=1} b_n x^n/n], or, exp[P.(b_1,...,b_n)x] = exp[-], expressed umbrally with <"power series"> denoting umbral evaluation (b.)^n = b_n within the power series. This e.g.f. is central to the paper by Maxim and Schuermannn on characteristic classes (cf. Friedrich and McKay also). - Tom Copeland, Nov 11 2015
The elementary Schur polynomials are given by S(n,x(1),x(2),...,x(n)) = P(n,x(1), 2*x(2),...,n*x(n)) / n!. See p. 12 of Carrell. - Tom Copeland, Feb 06 2016
These partition polynomials are also related to the Casimir invariants associated to quantum density states on p. 3 of Boya and Dixit and pp. 5 and 6 of Byrd and Khaneja. - Tom Copeland, Jul 24 2017
With the indeterminates (x_1,x_2,x_3,...) = (t,-c_2*t,-c_3*t,...) with c_n >0, umbrally P(n,a.) = P(n,t)|{t^n = a_n} = 0 and P(j,a.)P(k,a.) = P(j,t)P(k,t)|{t^n =a_n} = d_{j,k} >= 0 is the coefficient of x^j/j!*y^k/k! in the Taylor series expansion of the formal group law FGL(x,y) = f[f^{-1}(x)+f^{-1}(y)], where a_n are the inversion partition polynomials for calculating f(x) from the coefficients of the series expansion of f^{-1}(x) given in A133932. - Tom Copeland, Feb 09 2018
For relation to the Witt symmetric functions, as well as the basic power, elementary, and complete symmetric functions, see the Borger link p. 295. For relations to diverse zeta functions, determinants, and paths on graphs, see the MathOverflow question Cycling Through the Zeta Garden. - Tom Copeland, Mar 25 2018
Chmutov et al. identify the partition polynomials of this entry with the one-part Schur polynomials and assert that any linear combination with constant coefficients of these polynomials is a tau function for the KP hierarchy. - Tom Copeland, Apr 05 2018
With the indeterminates in the partition polynomials assigned as generalized harmonic numbers, i.e., as partial sums of the Dirichlet series for the Riemann zeta function, zeta(n), for integer n > 1, sums of simple normalizations of these polynomials give either unity or simple sums of consecutive zeta(n) (cf. Hoffman). Other identities involving these polynomials can be found in the Choi reference in Hoffman's paper. - Tom Copeland, Oct 05 2019
On p. 39 of Ma Luo's thesis is the e.g.f. of rational functions r_n obtained through the (umbral) formula 1/(1-r.T) = exp[log(1+P.T)], a differently signed e.g.f. of this entry, where (P.)^n = P_n are Eisenstein elliptic functions. P. 38 gives the example of 4! * r_4 as the signed 4th row partition polynomial of this entry. This series is equated through a simple proportionality factor to the Zagier Jacobi form on p. 25. Recurrence relations for the P_n are given on p. 24 involving the normalized k-weight Eisenstein series G_k introduced on p. 23 and related to the Bernoulli numbers. - Tom Copeland, Oct 16 2019
The Chern characteristic classes or forms of complex vector bundles and the characteristic polynomials of curvature forms for a smooth manifold can be expressed in terms of this entry's partition polynomials with the associated traces, or power sum polynomials, as the indeterminates. The Chern character is the e.g.f. of these traces and so its coefficients are given by the Faber polynomials with this entry's partition polynomials as the indeterminates. See the Mathoverflow question "A canonical reference for Chern characteristic classes". - Tom Copeland, Nov 04 2019
For an application to the physics of charged fermions in an external field, see Figueroa et al. - Tom Copeland, Dec 05 2019
Konopelchenko, in Proposition 5.2, p. 19, defines an operator P_k that is a differently signed operator version of the partition polynomials of this entry divided by a factorial. These operators give rise to bilinear Hirota equations for the KP hierarchy. These partition polynomials are also presented in Hopf algebras of symmetric functions by Cartier. - Tom Copeland, Dec 18 2019
For relationship of these partition polynomials to calculations of Pontryagin classes and the Riemann xi function, see A231846. - Tom Copeland, May 27 2020
Luest and Skliros summarize on p. 298 many of the properties of the cycle index polynomials given here; and Bianchi and Firrotta, a few on p. 6. - Tom Copeland, Oct 15 2020
From Tom Copeland, Oct 15 2020: (Start)
With a_n = n! * b_n = (n-1)! * c_n for n  > 0, represent a function with f(0) = a_0 = b_0 = 1 as an
A) exponential generating function (e.g.f), or formal Taylor series: f(x) = e^{a.x} = 1 + Sum_{n > 0} a_n * x^n/n!
B) ordinary generating function (o.g.f.), or formal power series: f(x)  = 1/(1-b.x) = 1 + Sum_{n > 0}  b_n * x^n
C) logarithmic generating function (l.g.f): f(x) = 1 - log(1 - c.x) = 1 + Sum_{n > 0}  c_n * x^n /n.
Expansions of log(f(x)) are given in
I) A127671 and A263634 for the e.g.f: log[ e^{a.*x} ] =  e^{L.(a_1,a_2,...)x} = Sum_{n > 0} L_n(a_1,...,a_n) * x^n/n!, the logarithmic polynomials, cumulant expansion polynomials
II) A263916 for the o.g.f.: log[ 1/(1-b.x) ] =  log[ 1 - F.(b_1,b_2,...)x ] = -Sum_{n > 0} F_n(b_1,...,b_n) * x^n/n, the Faber polynomials.
Expansions of exp(f(x)-1) are given in
III) A036040 for an e.g.f: exp[ e^{a.x} - 1 ] = e^{BELL.(a_1,...)x}, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind
IV) A130561 for an o.g.f.: exp[ b.x/(1-b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials
V) A036039 for an l.g.f.: exp[ -log(1-c.x) ] =  e^{CIP.(c_1,...)x}, the cycle index polynomials of the symmetric groups S_n, a.k.a. the Stirling partition polynomials of the first kind.
Since exp and log are a compositional inverse pair, one can extract the indeterminates of the log set of partition polynomials from the exp set and vice versa. For a discussion of the relations among these polynomials and the combinatorics of connected and disconnected graphs/maps, see Novak and LaCroix on classical moments and cumulants and the two books on statistical mechanics referenced in A036040. (End)

			The partition array T(n, k) begins (see the W. Lang link for rows 1..10):
  n\k   1    2    3    4    5    6    7    8    9   10   11  12   13  14 15 ...
  1:    1
  2:    1    1
  3:    2    3    1
  4:    6    8    3    6    1
  5:   24   30   20   20   15   10    1
  6:  120  144   90   40   90  120   15   40   45   15    1
  7:  720  840  504  420  504  630  280  210  210  420  105  70  105  21  1
... reformatted by _Wolfdieter Lang_, May 25 2019

Abramowitz and Stegun, Handbook, p. 831, column labeled "M_2".

David W. Wilson, Table of n, a(n) for n = 1..11731 (rows 1 through 26).
Milton Abramowitz and Irene A. Stegun, editors, Multinomials: M_1, M_2 and M_3, Handbook of Mathematical Functions, December 1972, pp. 831-2.
H. Airault and A. Bouali, Differential calculus on the Faber polynomials, Bulletin des Sciences Mathématiques, Volume 130, Issue 3, April-May 2006, Pages 179-222.
M. Bianchi and M. Firrotta, DDF operators, open string coherent states and their scattering amplitudes, arXiv:1902.07016 [hep-th], 2019.
J. Borger, Witt vectors, semirings, and total positivity, arXiv:1310.3013 [math.CO], 2015.
A. Bouali, Faber polynomials Cayley-Hamilton equation and Newton symmetric functions, Bulletin des Sciences Mathématiques, Volume 130, Issue 1, Jan-Feb 2006, Pages 49-70.
L. Boya and K. Dixit, Geometry of density states, arXiv:808.1930 [quant-phy], 2017.
M. Byrd and N. Khaneja, Characterization of the positivity of the density matrix in terms of the coherence vector representation , arXiv:quant-phy/0302024, 2003.
S. Carrell, Combinatorics of the KP Hierarchy, Thesis, University of Waterloo, Ontario, Canada, 2009.
P. Cartier, A primer of Hopf algebras, preprint, Institut des Hautes Etudes Scientifiques, France, 2006, pp. 56 and 57.
S. Chmutov, M. Kazarian, and S. Lando, Polynomial graph invariants and the KP hierarchy , arXiv:1803.09800 [math.CO], p. 16-17, 2018.
Tom Copeland, Lagrange a la Lah, 2011; Riemann zeta function at positive integers and an Appell sequence of polynomials, 2012; The creation / raising operators for Appell sequences, 2015; Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015; Addendum to Elliptic Lie Triad, 2015; Formal group laws and binomial Sheffer polynomials, 2018; Appells and Roses: Newton, Leibniz, Euler, Riemann and Symmetric Polynomials, 2020.
Mark Dominus, Cycle classes of permutations [From _Wouter Meeussen_, Jun 26 2009]
G. Duchamp, Important formulas in combinatorics: The exponential formula, a Mathoverflow answer, 2015.
H. Figueroa and J. Gracia-Bondia, Combinatorial Hopf algebras in quantum field theory I, arXiv:0408145 [hep-th], 2005, (pp. 40-41).
R. Friedrich and J. McKay, Formal groups, Witt vectors and free probability, arXiv:1204.6522 [math.OA], 2012.
M. Hoffman, Harmonic-number summation identities, symmetric functions, and multiple zeta values, arXiv:1602.03198 [math.NT], p. 2-3, 2016.
B. Konopelchenko, Quantum deformations of associative algebras and integrable systems, arXiv:0802.3022 [nlin.SI], 2008.
Wolfdieter Lang, First ten rows and polynomials.
D. Luest and D. Skliros, Handle Operators in String Theory, arXiv:1912.01055 [hep-th], 2019.
Ma Luo, Algebraic De Rham Theory for Completions of Fundamental Groups of Moduli Spaces of Elliptic Curves, Ph.D. dissertation, Math Dept., Duke Univ., 2018.
Ma Luo, Algebraic De Rham Theory for Completions of Fundamental Groups of Moduli Spaces of Elliptic Curves, arXiv:1804.06977 [math.NT], 2019.
MathOverflow, Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants, a question posed by Tom Copeland, 2012.
MathOverflow, Canonical reference for Chern characteristic classes, a question posed by Tom Copeland, 2019.
L. Maxim and J. Schuermann, Equivariant characteristic classes of external and symmetric products of varieties, arXiv:1508.04356 [math.AG], 2015.
R. Szabo, N=2 gauge theories, instanton moduli spaces and geometric representation theory, arXiv:1507.00685 [hep-th], 2015.
Andrei Vieru, Analytic renormalization of multiple zeta functions. Geometry and combinatorics of generalized Euler reflection formula for MZV, arXiv preprint arXiv:1601.04703 [math.NT], 2016.

Cf. A036036, A036037, A036038, A036040.
Cf. other versions based on different partition orderings: A102189 (rows reversed), A181897, A319192.
Cf. A094587, A115131, A130561, A263634, A263916.
Cf. A133932.
Cf. A231846.
Cf. A127671.

nmax:=7: 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 sA036039(n, m) := n!/ (mul((t)^q(t)*q(t)!, t=1..n)); od: od: seq(seq(A036039(n, m), m=1..numbpart(n)), n=1..nmax); # Johannes W. Meijer, Jul 14 2016
# 2nd program:
A036039 := proc(n,k)
    local a,prts,e,ai ;
    a := n! ;
    # ASPrts is implemented in A119441
    prts := ASPrts(n)[k] ;
    ai := 1;
    for e from 1 to nops(prts) do
        if e>1 then
            if op(e,prts) = op(e-1,prts) then
                ai := ai+1 ;
            else
                ai := 1;
            end if;
        end if;
        a := a/(op(e,prts)*ai) ;
    end do:
    a ;
end proc:
seq(seq(A036039(n,k),k=1..combinat[numbpart](n)),n=1..15) ; # R. J. Mathar, Dec 18 2016

aspartitions[n_]:=Reverse/@Sort[Sort/@IntegerPartitions[n]];(* Abramowitz & Stegun ordering *);
ascycleclasses[n_Integer]:=n!/(Times@@ #)&/@((#!
Range[n]^#)&/@Function[par,Count[par,# ]&/@Range[n]]/@aspartitions[n])
(* The function "ascycleclasses" is then identical with A&S multinomial M2. *)
Table[ascycleclasses[n], {n, 1, 8}] // Flatten
(* Wouter Meeussen, Jun 26 2009, Jun 27 2009 *)

def PartAS(n):
    P = []
    for k in (1..n):
        Q = [p.to_list() for p in Partitions(n, length=k)]
        for q in Q: q.reverse()
        P = P + sorted(Q)
    return P
def A036039_row(n):
    fn, C = factorial(n), []
    for q in PartAS(n):
        q.reverse()
        p = Partition(q)
        fp = 1; pf = 1
        for a, c in p.to_exp_dict().items():
            fp *= factorial(c)
            pf *= factorial(a)**c
        co = fn//(fp*pf)
        C.append(co*prod([factorial(i-1) for i in p]))
    return C
for n in (1..10):
    print(A036039_row(n)) # Peter Luschny, Dec 18 2016

T(n,k) = n!/Product_{j=1..n} j^a(n,k,j)*a(n,k,j)!, with the k-th partition of  n >= 1 in Abromowitz-Stegun order written as Product_{j=1..n} j^a(n,k,j) with nonnegative integers a(n,k,j) satisfying Sum_{j=1..n} j*a(n,k,j) = n, and the number of parts is Sum_{j=1..n} a(n,k,j) =: m(n,k). - Wolfdieter Lang, May 25 2019
Raising and lowering operators are given for the partition polynomials formed from this sequence in the link in "Lagrange a la Lah Part I" on p. 23. - Tom Copeland, Sep 18 2011
From Szabo p. 34, with b_n = q^n / (1-q^n)^2, the partition polynomials give an expansion of the MacMahon function M(q) = Product_{n>=1} 1/(1-q^n)^n = Sum_{n>=0} PL(n) q^n, the generating function for PL(n) = n! P_n(b_1,...,b_n), the number of plane partitions with sum n. - Tom Copeland, Nov 11 2015
From Tom Copeland, Nov 18 2015: (Start)
The partition polynomials of A036040 are obtained by substituting x[n]/(n-1)! for x[n] in the partition polynomials of this entry.
CIP_n(t-F(1,b1),-F(2,b1,b2),...,-F(n,b1,...,bn)) = P_n(b1,...,bn;t), where CIP_n are the partition polynomials of this entry; F(n,...), those of A263916; and P_n, those defined in my formula in A094587, e.g., P_2(b1,b2;t) = 2 b2 + 2 b1 t + t^2.
CIP_n(-F(1,b1),-F(2,b1,b2),...,-F(n,b1,...,bn)) = n! bn. (End)
From the relation to the elementary Schur polynomials given in A130561 and above, the partition polynomials of this array satisfy (d/d(x_m)) P(n,x_1,...,x_n) = (1/m) * (n!/(n-m)!) * P(n-m,x_1,...,x_(n-m)) with P(k,...) = 0 for k<0. - Tom Copeland, Sep 07 2016
Regarded as Appell polynomials in the indeterminate x(1)=u, the partition polynomials of this entry P_n(u) obey d/du P_n(u) = n * P_{n-1}(u), so the abscissas for the zeros of P_n(u) are the same as those of the extrema of P{n+1}(u). In addition, the coefficient of u^{n-1} in P_{n}(u) is zero since these polynomials are related to the characteristic polynomials of matrices with null main diagonals, and, therefore, the trace is zero, further implying the abscissa for any zero is the negative of the sum of the abscissas of the remaining zeros. This assumes all zeros are distinct and real. - Tom Copeland, Nov 10 2019

More terms from David W. Wilson

Title expanded by Tom Copeland, Oct 15 2020

A036037 Triangle read by rows in which row n lists all the parts of all the partitions of n, sorted first by length and then colexicographically.

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

Offset: 1

N. J. A. Sloane

First differs from A334439 for partitions of 9. Namely, this sequence has (4,4,1) before (5,2,2), while A334439 has (5,2,2) before (4,4,1). - Gus Wiseman, May 08 2020
This is also a list of all the possible prime signatures of a number, arranged in graded colexicographic ordering. - N. J. A. Sloane, Feb 09 2014
This is also the Abramowitz-Stegun ordering of reversed partitions (A036036) if the partitions are reversed again after sorting. Partitions sorted first by sum and then colexicographically are A211992. - Gus Wiseman, May 08 2020

			First five rows are:
{{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}}
Up to the fifth row, this is exactly the same as the reverse lexicographic ordering A080577. The first row which differs is the sixth one, which reads ((6), (5,1), (4,2), (3,3), (4,1,1), (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)). - _M. F. Hasler_, Jan 23 2020
From _Gus Wiseman_, May 08 2020: (Start)
The sequence of all partitions begins:
  ()         (3,2)        (2,1,1,1,1)
  (1)        (3,1,1)      (1,1,1,1,1,1)
  (2)        (2,2,1)      (7)
  (1,1)      (2,1,1,1)    (6,1)
  (3)        (1,1,1,1,1)  (5,2)
  (2,1)      (6)          (4,3)
  (1,1,1)    (5,1)        (5,1,1)
  (4)        (4,2)        (4,2,1)
  (3,1)      (3,3)        (3,3,1)
  (2,2)      (4,1,1)      (3,2,2)
  (2,1,1)    (3,2,1)      (4,1,1,1)
  (1,1,1,1)  (2,2,2)      (3,2,1,1)
  (5)        (3,1,1,1)    (2,2,2,1)
  (4,1)      (2,2,1,1)    (3,1,1,1,1)
(End)

Robert Price, Table of n, a(n) for n = 1..3615, 15 rows.
Wikiversity, Lexicographic and colexicographic order

See A036036 for the graded reflected colexicographic ("Abramowitz and Stegun" or Hindenburg) ordering.
See A080576 for the graded reflected lexicographic ("Maple") ordering.
See A080577 for the graded reverse lexicographic ("Mathematica") ordering: differs from a(48) on!
See A228100 for the Fenner-Loizou (binary tree) ordering.
See also A036038, A036039, A036040: (multinomial coefficients).
Partition lengths are A036043.
Reversing all partitions gives A036036.
The number of distinct parts is A103921.
Taking Heinz numbers gives A185974.
The version ignoring length is A211992.
The version for revlex instead of colex is A334439.
Lexicographically ordered reversed partitions are A026791.
Reverse-lexicographically ordered partitions are A080577.
Sorting partitions by Heinz number gives A296150.
Cf. A000041, A124734, A193073, A228100, A228531, A296774, A334301, A334433, A334436, A334437, A334442.

Reverse/@Join@@Table[Sort[Reverse/@IntegerPartitions[n]],{n,8}] (* Gus Wiseman, May 08 2020 *)
- or -
colen[f_,c_]:=OrderedQ[{Reverse[f],Reverse[c]}];
Join@@Table[Sort[IntegerPartitions[n],colen],{n,8}] (* Gus Wiseman, May 08 2020 *)

Name corrected by Gus Wiseman, May 12 2020

Mathematica programs corrected to reflect offset of one and not zero by Robert Price, Jun 04 2020

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

A193073 Triangle in which n-th row lists all partitions of n, in graded lexicographical ordering.

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

Offset: 1

M. F. Hasler, Jul 15 2011

The partitions of the integer n are sorted in lexicographical order (cf. link: sums are written with terms in decreasing order, then they are sorted in lexicographical (increasing) order), i.e., as [1,1,...,1], [2,1,...,1], [2,2,...], ..., [n].

			First five rows are:
[[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]]
From _Gus Wiseman_, May 08 2020: (Start)
The sequence of all partitions begins:
  ()           (2,2,1)        (5,1)            (5,2)
  (1)          (3,1,1)        (6)              (6,1)
  (1,1)        (3,2)          (1,1,1,1,1,1,1)  (7)
  (2)          (4,1)          (2,1,1,1,1,1)    (1,1,1,1,1,1,1,1)
  (1,1,1)      (5)            (2,2,1,1,1)      (2,1,1,1,1,1,1)
  (2,1)        (1,1,1,1,1,1)  (2,2,2,1)        (2,2,1,1,1,1)
  (3)          (2,1,1,1,1)    (3,1,1,1,1)      (2,2,2,1,1)
  (1,1,1,1)    (2,2,1,1)      (3,2,1,1)        (2,2,2,2)
  (2,1,1)      (2,2,2)        (3,2,2)          (3,1,1,1,1,1)
  (2,2)        (3,1,1,1)      (3,3,1)          (3,2,1,1,1)
  (3,1)        (3,2,1)        (4,1,1,1)        (3,2,2,1)
  (4)          (3,3)          (4,2,1)          (3,3,1,1)
  (1,1,1,1,1)  (4,1,1)        (4,3)            (3,3,2)
  (2,1,1,1)    (4,2)          (5,1,1)          (4,1,1,1,1)
The triangle with partitions shown as Heinz numbers (A334434) begins:
    1
    2
    4   3
    8   6   5
   16  12   9  10   7
   32  24  18  20  15  14  11
   64  48  36  27  40  30  25  28  21  22  13
  128  96  72  54  80  60  45  50  56  42  35  44  33  26  17
(End)

Alois P. Heinz, Rows n = 1..19, flattened
OEIS Wiki, Orderings of partitions.
Wikiversity, Lexicographic and colexicographic order

See A036036 for the Hindenburg (graded reflected colexicographic) ordering (listed in the Abramowitz and Stegun Handbook).
See A036037 for graded colexicographic ordering.
See A080576 for the Maple (graded reflected lexicographic) ordering.
See A080577 for the Mathematica (graded reverse lexicographic) ordering.
See A228100 for the Fenner-Loizou (binary tree) ordering.
A006128 gives row lengths.
Row n has A000041(n) partitions.
The version for reversed (weakly increasing) partitions is A026791.
Lengths of these partitions appear to be A049085.
Taking colex instead of lex gives A211992.
The generalization to compositions is A228351.
Sorting partitions by Heinz number gives A296150.
The length-sensitive refinement is A334301.
The Heinz numbers of these partitions are A334434.
Cf. A066099, A129129, A185974, A228531, A334302, A334433, A334437, A334439.

row[n_] := Flatten[Reverse[Reverse /@ SplitBy[IntegerPartitions[n], Length] ], 1]; Array[row, 19] // Flatten (* Jean-François Alcover, Dec 05 2016 *)
lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
Join@@Table[Sort[IntegerPartitions[n],lexsort],{n,0,8}] (* Gus Wiseman, May 08 2020 *)

A193073_row(n)=concat(vecsort(apply(P->Vec(vecsort(P,,4)),partitions(n)))) \\ The two vecsort() are needed since the PARI function (version >= 2.7.1) yields the partitions in Abramowitz-Stegun order: sorted by increasing length, decreasing largest part, then lex order, with parts in increasing order. - M. F. Hasler, Jun 04 2018 [replaced older code from Jul 12 2015]

def p(n, i):
    if n==0 or i==1: return [[1]*n]
    T = [[i] + x for x in p(n-i, i)] if i<=n else []
    return p(n, i-1) + T
A193073 = lambda n: p(n,n)
for n in (1..5): print(A193073(n)) # Peter Luschny, Aug 07 2015

A036043 Irregular triangle read by rows: row n (n >= 0) gives number of parts in all partitions of n (in Abramowitz and Stegun 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, 2, 3, 3, 3, 4, 4, 5, 6, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7, 8, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 8, 9

Offset: 0

N. J. A. Sloane

The sequence of row lengths of this array is p(n) = A000041(n) (partition numbers).
The sequence of row sums is A006128(n).
The number of times k appears in row n is A008284(n,k). - Franklin T. Adams-Watters, Jan 12 2006
The next level (row) 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. See A128628. - Alford Arnold, Mar 27 2007
The 1's in the (flattened) sequence mark the start of a new row, the value that precedes the 1 equals the row number minus one. (I.e., the 1 preceded by a 0 is the start of row 1, the 1 preceded by a 6 is the start of row 7, etc.) - M. F. Hasler, Jun 06 2018
Also the maximum part in the n-th partition in graded lexicographic order (sum/lex, A193073). - Gus Wiseman, May 24 2020

			0;
1;
1, 2;
1, 2, 3;
1, 2, 2, 3, 4;
1, 2, 2, 3, 3, 4, 5;
1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6;
1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7;

Abramowitz and Stegun, Handbook, p. 831, column labeled "m".

T. D. Noe, Rows n = 0..25 of irregular triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [scanned copy], p. 831.
Kevin Brown, Generalized Birthday Problem (N Items in M Bins), 1994-2010.
Wolfdieter Lang, Rows n = 1 ..20.
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Cf. A036037, A036038, A036039, A036040, A036042.
Row lengths are A000041.
Partition lengths of A036036 and A334301.
The version not sorted by length is A049085.
The generalization to compositions is A124736.
The Heinz number of the same partition is A334433.
The number of distinct elements in the same partition is A334440.
The maximum part of the same partition is A334441.
Lexicographically ordered reversed partitions are A026791.
Lexicographically ordered partitions are A193073.
Cf. A103921, A115623, A124734, A185974, A296150, A334435, A334438, A334439.

with(combinat): nmax:=9: for n from 1 to nmax do y(n):=numbpart(n): P(n):=sort(partition(n)): for k from 1 to y(n) do B(k) := P(n)[k] od: for k from 1 to y(n) do s:=0: j:=0: while sJohannes W. Meijer, Jun 21 2010, revised Nov 29 2012
# alternative implementation based on A119441 by R. J. Mathar, Jul 12 2013
A036043 := proc(n,k)
    local pi;
    pi := ASPrts(n)[k] ;
    nops(pi) ;
end proc:
for n from 1 to 10 do
    for k from 1 to A000041(n) do
        printf("%d,",A036043(n,k)) ;
    end do:
    printf("\n") ;
end do:

Table[Length/@Sort[IntegerPartitions[n]],{n,0,30}] (* Gus Wiseman, May 22 2020 *)

A036043(n,k)=#partitions(n)[k] \\ M. F. Hasler, Jun 06 2018

def A036043_row(n):
    return [len(p) for k in (0..n) for p in Partitions(n, length=k)]
for n in (0..10): print(A036043_row(n)) # Peter Luschny, Nov 02 2019

a(n) = A001222(A334433(n)). - Gus Wiseman, May 22 2020

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001
a(0) inserted by Franklin T. Adams-Watters, Jun 24 2014
Incorrect formula deleted by M. F. Hasler, Jun 06 2018

A124734 Table with all compositions sorted first by total, then by length and finally lexicographically.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Nov 06 2006

This is similar to the Abramowitz and Stegun ordering for partitions (see A036036). The standard ordering for compositions is A066099, which is more similar to the Mathematica partition ordering (A080577).
This can be regarded as a table in two ways: with each composition as a row, or with the compositions of each integer as a row. The first way has A124736 as row lengths and A070939 as row sums; the second has A001792 as row lengths and A001788 as row sums.
This sequence includes every finite sequence of positive integers.

			The table starts:
1
2; 1 1
3; 1 2; 2 1; 1 1 1
4; 1 3; 2 2; 3 1; 1 1 2; 1 2 1; 2 1 1; 1 1 1 1;

Alois P. Heinz, Rows n = 1..11, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A001788, A001792, A036036, A066099, A070939, A080577, A124735, A124736.

Table[Sort@Flatten[Permutations /@ IntegerPartitions@n, 1], {n, 8}] // Flatten (* Robert Price, Jun 13 2020 *)

A026792 List of juxtaposed reverse-lexicographically ordered partitions of the positive integers.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

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 (list-)reversed lexicographic; see examples. [Joerg Arndt, Sep 03 2013]
Written as a triangle; row n has length A006128(n); row sums give A066186. Also written as an irregular tetrahedron in which T(n,j,k) is the k-th largest part of the j-th partition of n; the sum of column k in the slice n is A181187(n,k); right border of the slices gives A182715. - Omar E. Pol, Mar 25 2012
The equivalent sequence for compositions (ordered partitions) is A228351. - Omar E. Pol, Sep 03 2013
This is the reverse-colexicographic order of integer partitions, or the reflected reverse-lexicographic order of reversed integer partitions. It is not reverse-lexicographic order (A080577), wherein we would have (3,1) before (2,2). - Gus Wiseman, May 12 2020

			E.g. the partitions of 3 (3,2+1,1+1+1) appear as the string 3,2,1,1,1,1.
So the list begins:
1
2, 1, 1,
3, 2, 1, 1, 1, 1,
4, 2, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1,
5, 3, 2, 4, 1, 2, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1,
...
From _Omar E. Pol_, Sep 03 2013: (Start)
Illustration of initial terms:
---------------------------------
n  j     Diagram     Partition
---------------------------------
.         _
1  1     |_|         1;
.         _ _
2  1     |_  |       2,
2  2     |_|_|       1, 1;
.         _ _ _
3  1     |_ _  |     3,
3  2     |_  | |     2, 1,
3  3     |_|_|_|     1, 1, 1;
.         _ _ _ _
4  1     |_ _    |   4,
4  2     |_ _|_  |   2, 2,
4  3     |_ _  | |   3, 1,
4  4     |_  | | |   2, 1, 1,
4  5     |_|_|_|_|   1, 1, 1, 1;
...
(End)
From _Gus Wiseman_, May 12 2020: (Start)
This sequence can also be interpreted as the following triangle, whose n-th row is itself a finite triangle with A000041(n) rows. Showing these partitions as their Heinz numbers gives A334436.
                             0
                            (1)
                          (2)(11)
                        (3)(21)(111)
                   (4)(22)(31)(211)(1111)
             (5)(32)(41)(221)(311)(2111)(11111)
  (6)(33)(42)(222)(51)(321)(411)(2211)(3111)(21111)(111111)
(End)

Robert Price, Table of n, a(n) for n = 1..3615, 15 rows.
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Cf. A026791, A228531.
The reflected version for reversed partitions is A080577.
The partition minima appear to be A182715.
The graded reversed version is A211992.
The version for compositions is A228351.
The Heinz numbers of these partitions are A334436.
Cf. A000041, A036036, A036037, A193073, A296150, A331581, A334301, A334435, A334437, A334439.

revcolex[f_,c_]:=OrderedQ[PadRight[{Reverse[c],Reverse[f]}]];
Join@@Table[Sort[IntegerPartitions[n],revcolex],{n,0,8}] (* reverse-colexicographic order, Gus Wiseman, May 10 2020 *)
- or -
revlex[f_,c_]:=OrderedQ[PadRight[{c,f}]];
Reverse/@Join@@Table[Sort[Reverse/@IntegerPartitions[n],revlex],{n,0,8}] (* reflected reverse-lexicographic order, Gus Wiseman, May 12 2020 *)

Terms 81st, 83rd and 84th corrected by Omar E. Pol, Aug 16 2009

cp's OEIS Frontend

A036038 Triangle of multinomial coefficients.

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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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A036039 Irregular triangle of multinomial coefficients of integer partitions read by rows (in Abramowitz and Stegun ordering) giving the coefficients of the cycle index polynomials for the symmetric groups S_n.

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A036037 Triangle read by rows in which row n lists all the parts of all the partitions of n, sorted first by length and then colexicographically.

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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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A193073 Triangle in which n-th row lists all partitions of n, in graded lexicographical ordering.

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A036043 Irregular triangle read by rows: row n (n >= 0) gives number of parts in all partitions of n (in Abramowitz and Stegun order).

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