A036043 -id:A036043 - cp's OEIS frontend

A115131 Waring numbers for power sums functions in terms of elementary symmetric functions; irregular triangle T(n,k), read by rows, for n >= 1 and 1 <= k <= A000041(n).

1, -2, 1, 3, -3, 1, -4, 4, 2, -4, 1, 5, -5, -5, 5, 5, -5, 1, -6, 6, 6, 3, -6, -12, -2, 6, 9, -6, 1, 7, -7, -7, -7, 7, 14, 7, 7, -7, -21, -7, 7, 14, -7, 1, -8, 8, 8, 8, 4, -8, -16, -16, -8, -8, 8, 24, 12, 24, 2, -8, -32, -16, 8, 20, -8, 1, 9, -9, -9, -9, -9, 9, 18, 18, 9, 9, 18, 3, -9, -27, -27, -27, -27, -9, 9, 36, 18, 54, 9, -9, -45, -30, 9, 27, -9, 1

Offset: 1

Wolfdieter Lang, Jan 13 2006

N*t^{(N)}n(sigma_1, ..., sigma_N):= sum((x_k)^n, k=1..N) with the elementary symmetric function sigma_k (superscript (N) omitted) in terms of the indeterminates x_1,...,x_N, is an N-variable generalization of Chebyshev's polynomials C_n((sigma_1)/2) = t^{(N=2)}_n(sigma_1, sigma_2 = 1). In general, C_n^{(N)}(sigma_1, ..., sigma{N-1}) := t^{(N)}n(sigma_1, ..., sigma{N-1}, sigma_N:=1). If n > N, one puts sigma_{N+1} = 0, ..., sigma_n = 0.
The sequence of row lengths of this array is A000041(n) (partition numbers).
In row n, this triangular array uses partitions of n listed in the Abramowitz-Stegun order (compare with the M_0, M_1, M_2 and M_3 numbers given in A048996 = |A111786|, A036038, A036039 and A036040, resp.).
Row sums give (-1)^(n-1). Unsigned row sums give A000225(n)= 2^n - 1.
N*t^{(N)}n(sigma_1, ..., sigma_N) gives the sum of the n-th power of the indeterminates x_1, ... , x_N in terms of the elementary symmetric functions of these indeterminates. The coefficient T(n, k) of this partition array corresponds to the k-th partition of n in the Abramowitz-Stegun order, and it multiplies the product of sigma_j functions encoded by this partition. See the example for n = 4 below. - _Wolfdieter Lang, Mar 09 2015

			First few rows of triangle T(n,k) are as follows (see the link for rows 1..10):
   1;
  -2,  1;
   3, -3,  1;
  -4,  4,  2, -4, 1;
   5, -5, -5,  5, 5, -5, 1;
  ...
n=4: N*t^{(N)}_4 = -4*(sigma_4)^1 + 4*(sigma_1)*(sigma_3) + 2*(sigma_2)^2 -4*(sigma_1)^2*(sigma_2) + 1*(sigma_1)^4.
  (For 2 <= N < 4, one puts sigma_{N+1} = 0 = ... = sigma_4 = 0.) This becomes Sum_{k = 1..N} (x_k)^4 if the sigma functions are written in terms of the variables x_1, x_2, ..., x_N. E.g., for N=2: 0 + 0 + 2*(x_1*x_2)^2 -4*(x_1 + x_2)^2*(x_1*x_2) + 1*(x_1 + x_2)^4 = (x_1)^4 + (x_2)^4.

P. A. MacMahon, Combinatory Analysis, 2 vols., Chelsea, NY, 1960, see p. 5 (with a_k -> sigma_k).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972; see pp. 831-832. [alternative scanned copy].
Wolfdieter Lang, On sums of powers of zeros of polynomials, J. Comp. Appl. Math. 89 (1998) 237-256; see Theorem 1.
Wolfdieter Lang, First 10 rows of the array.
R. Lidl, Tschebyscheffpolynome in mehreren Variablen, J. reine u. angew. Math. 273 (1975), 178-198.
R. Lidl, Tschebyscheffpolynome in mehreren Variablen, J. reine u. angew. Math. 273 (1975), 178-198.
R. Lidl and Ch. Wells, Chebyshev polynomials in several variables, J. reine u. angew. Math. 255 (1972), 104-111.
R. Lidl and Ch. Wells, Chebyshev polynomials in several variables, J. reine u. angew. Math. 255 (1972), 104-111.
P. A. MacMahon, Combinatory analysis (2 vols.), Chelsea, NY, 1960; see p. 5 (with a_k -> sigma_k).

Cf. A000041, A000225, A036038, A036039, A036040, A048996, A111786.
Cf. A210258 (in another ordering of partitions), A132460 (N=2), A325477 (N=3),
A324602 (N=4).

T(n,k) = (n/m(n,k))*A111786(n,k) for the k-th partition of n with m(n,k) parts in the Abramowitz-Stegun order for n >= 1 and k = 1..p(n), where p(n) := A000041(n).

Explicitly: T(n,k) = (-1)^(n + m(n,k)) * n * (m(n,k) - 1)!/(Product_{j = 1..n} e(k,j)!), where m(n,k):= Sum_{j = 1..n} e(k,j), with [1^e(k, 1), 2^e(k,2), ..., n^e(k,n)] being the k-th partition of n in the mentioned order. For m(n,k), see A036043.

Various sections edited by Petros Hadjicostas, Dec 14 2019

A035206 Number of multisets associated with least integer of each prime signature.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 4, 12, 6, 12, 1, 5, 20, 20, 30, 30, 20, 1, 6, 30, 30, 15, 60, 120, 20, 60, 90, 30, 1, 7, 42, 42, 42, 105, 210, 105, 105, 140, 420, 140, 105, 210, 42, 1, 8, 56, 56, 56, 28, 168, 336, 336, 168, 168, 280, 840, 420, 840, 70, 280, 1120, 560, 168, 420, 56, 1, 9, 72

Offset: 0

Alford Arnold

a(n,k) multiplied by A036038(n,k) yields A049009(n,k).
a(n,k) enumerates distributions of n identical objects (balls) into m of altogether n distinguishable boxes. The k-th partition of n, taken in the Abramowitz-Stegun (A-St) order, specifies the occupation of the m =m(n,k)= A036043(n,k) boxes. m = m(n,k) is the number of parts of the k-th partition of n. For the A-St ordering see pp.831-2 of the reference given in A117506. - Wolfdieter Lang, Nov 13 2007
The sequence of row lengths is p(n)= A000041(n) (partition numbers).
For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
The corresponding triangle with summed row entries which belong to partitions of the same number of parts k is A103371. [Wolfdieter Lang, Jul 11 2012]

			n\k 1  2  3  4   5   6   7   8   9  10  11  12  13 14 15
0   1
1   1
2   2  1
3   3  6  1
4   4 12  6 12   1
5   5 20 20 30  30  20   1
6   6 30 30 15  60 120  20  60  90  30   1
7   7 42 42 42 105 210 105 105 140 420 140 105 210 42  1
...
Row No. 8:  8  56 56 56 28 168 336 336 168 168 280  840 420 840 70 280 1120 560 168 420 56 1
Row No. 9: 9 72 72 72 72 252 504 504 252 252 504 84 504 1512 1512 1512 1512 504 630 2520 1260 3780 630 504 2520 1680 252 756 72 1
[rewritten and extended table by _Wolfdieter Lang_, Jul 11 2012]
a(5,5) relates to the partition (1,2^2) of n=5. Here m=3 and 5 indistinguishable (identical) balls are put into boxes b1,...,b5 with m=3 boxes occupied; one with one ball and two with two balls.
Therefore a(5,5) = binomial(5,3)*3!/(1!*2!) = 10*3 = 30. _Wolfdieter Lang_, Nov 13 2007

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, Tenth Printing, 1972 [alternative scanned copy].
Wolfdieter Lang, First 10 rows and more.

Cf. A036038, A048996, A049009.
Cf. A001700 (row sums).
Cf. A103371(n-1, m-1) (triangle obtained after summing in every row the numbers with like part numbers m).

C(sig)={my(S=Set(sig)); binomial(vecsum(sig), #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

a(n,k) = A048996(n,k)*binomial(n,m(n,k)),n>=1, k=1,...,p(n) and m(n,k):=A036043(n,k) gives the number of parts of the k-th partition of n.

More terms from Joshua Zucker, Jul 27 2006

a(0)=1 prepended by Andrew Howroyd, Oct 18 2020

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)).

A111785 T(n,k) are coefficients used for power series inversion (sometimes called reversion), n >= 0, k = 1..A000041(n), read by rows.

Original entry on oeis.org

1, -1, -1, 2, -1, 5, -5, -1, 6, 3, -21, 14, -1, 7, 7, -28, -28, 84, -42, -1, 8, 8, 4, -36, -72, -12, 120, 180, -330, 132, -1, 9, 9, 9, -45, -90, -45, -45, 165, 495, 165, -495, -990, 1287, -429, -1, 10, 10, 10, 5, -55, -110, -110, -55, -55, 220, 660, 330, 660, 55, -715, -2860, -1430, 2002, 5005, -5005, 1430, -1, 11, 11

Offset: 0

Wolfdieter Lang, Aug 23 2005

Coefficients are listed in Abramowitz and Stegun order (A036036).
The formula for the inversion of the power series y = F(x) = x*G(x) = x*(1 + Sum_{k>=1} g[k]*(x^k)) is obtained as a corollary of Lagrange's inversion theorem. The result is F^{(-1)}(y)= Sum_{n>=1} P(n-1)*y^n, where P(n):=sum over partitions of n of a(n,k)* G[k], with G[k]:=g[1]^e(k,1)*g[2]^e(k,2)*...*g[n]^e(k,n) if the k-th partition of n, in Abramowitz-Stegun order(see the given ref, pp. 831-2), is [1^e(k,1),2^e(k,2),...,n^e(k,n)], for k=1..p(n):= A000041(n) (partition numbers).
The sequence of row lengths is A000041(n) (partition numbers).
The signs are given by (-1)^m(n,k), with the number of parts m(n,k) = Sum_{j=1..n} e(k,j) of the k-th partition of n. For m(n,k) see A036043.
The proof that the unsigned row sums give Schroeder's little numbers A001003(n) results from their formula ((d^(n-1)/dx^(n-1)) ((1-x)/(1-2*x))^n)/n!|_{x=0}, n >= 1. This formula for A001003 can be proved starting with the compositional inverse of the g.f. of A001003 (which is given there in a comment) and using Lagrange's inversion theorem to recover the original sequence A001003.
For alternate formulations and relation to the geometry of associahedra or Stasheff polytopes (and other combinatorial objects) see A133437. [Tom Copeland, Sep 29 2008]
The coefficients of the row polynomials P(n) with monomials in lexicographically descending order e.g. P(6) = -1*g[6] + 8*g[5]*g[1] + 8*g[4]*g[2] - 36*g[4]*g[1]^2 + 4*g[3]^2 - 72*g[3]*g[2]*g[1] - 12*g[2]^3 + 120*g[3]*g[1]^3 + 180*g[2]^2*g[1]^2 - 330*g[2]*g[1]^4 + 132*g[1]^6 are given in A304462. [Herbert Eberle, Aug 16 2018]

			[ +1];
[ -1];
[ -1, 2];
[ -1, 5, -5];
[ -1, 6,  3, -21,  14];
[ -1, 7,  7, -28, -28, 84, -42];
[ -1, 8,  8,   4, -36, -72, -12, 120, 180, -330, 132];  ...
The seventh row, [ -1, 8, 8, 4, -36, -72, -12, 120, 180, -330, 132], stands for the row polynomial P(6) with monomials in lexicographically ascending order P(6) = -1*g[0]^5*g[6] + 8*g[0]^4*g[1]*g[5] + 8*g[0]^4*g[2]*g[4] + 4*g[0]^4*g[3]^2 - 36*g[0]^3*g[1]^2*g[4] - 72*g[0]^3*g[1]*g[2]*g[3] - 12*g[0]^3*g[2]^3 + 120*g[0]^2*g[1]^3*g[3] + 180*g[0]^2*g[1]^2*g[2]^2 - 330*g[0]*g[1]^4*g[2] + 132*g[1]^6 = (1/7!)*(differentiate 1/G(x)^7 six times and evaluate at x = 0). This gives the coefficient of y^7 of F^{(-1)}(y).

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 150, Table 4.1 (unsigned).

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, Tenth Printing, 1972 [alternative scanned copy].
A. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 16, 3.6.25.
Bartomeu Fiol and Alan Rios Fukelman, On the planar free energy of matrix models, arXiv:2111.14783 [hep-th], 2021. See also J. High Energy Phys. (2022) Iss. 2.
Wolfdieter Lang, First 10 rows and a formula.
Jin Wang, Nonlinear Inverse Relations for Bell Polynomials via the Lagrange Inversion Formula, J. Int. Seq., Vol. 22 (2019), Article 19.3.8.
Eric Weisstein's World of Mathematics, Series Reversion

Row sums give (-1)^n. Unsigned row sums are A001003(n) (little Schroeder numbers). Inversion triangle with leading quadratic term: A276738. Conjectured simplification: A283298.

(* Graded Colex Ordering: by length, then reverse lexicographic by digit *)
ClearAll[P, L, T, c, g]
P[0] := 1
P[n_] := -Total[
   Multinomial @@ # c[Total@# - 1] Times @@
       Power[g[#] & /@ Range[0, n - 1], #] & /@
    Table[ Count[p, i], {p, Drop[IntegerPartitions[n + 1], 1]}, {i,
      n}]]
L[n_] := Join @@ GatherBy[IntegerPartitions[n], Length]
T[1] := {1}
T[n_] := Coefficient[ Do[g[i] = P[i], {i, 0, n - 1}];
    P[n - 1], #] & /@ (Times @@@ Map[c, L[n - 1], {2}])
Array[T, 9] // Flatten (* Bradley Klee and Michael Somos, Apr 14 2017 *)

sv(n)={eval(Str("'s",n))}
Trm(q,v)={my(S=Set(v)); for(i=1, #S, my(x=S[i], c=#select(y->y==x, v)); q=polcoef(q, c, sv(x))); q}
Q(n)={polcoef(serreverse(x + x*sum(k=1, n, x^k*sv(k), O(x*x^n)))/x, n)}
row(n)={my(q=Q(n)); [Trm(q,Vec(v)) | v<-partitions(n)]} \\ Andrew Howroyd, Feb 01 2022

C(v)={my(n=vecsum(v), S=Set(v)); (-1)^#v*(n+#v)!/(n+1)!/prod(i=1, #S, my(x=S[i], c=#select(y->y==x, v)); c!)}
row(n)=[C(Vec(p)) | p<-partitions(n)]
{ for(n=0, 7, print(row(n))) } \\ Andrew Howroyd, Feb 01 2022

def A111785_list(dim): # returns the first dim rows
    C = [[0 for k in range(m+1)] for m in range(dim+1)]
    C[0][0] = 1; F = [1]; i = 1
    X = lambda n: 1 if n == 1 else var('x'+str(n))
    while i <= dim: F.append(F[i-1]*X(i)); i += 1
    for m in (1..dim):
        C[m][m] = -C[m-1][m-1]/F[1]
        for k in range(m-1, 0, -1):
            C[m][k] = -(C[m-1][k-1]+sum(F[i]*C[m][k+i-1] for i in (2..m-k+1)))/F[1]
    P = [expand((-1)^m*C[m][1]) for m in (1..dim)]
    R = PolynomialRing(ZZ, [X(i) for i in (2..dim)], order='lex')
    return [R(p).coefficients()[::-1] for p in P]
A111785_list(8) # Peter Luschny, Apr 14 2017

For row n >= 1 the row polynomial in the variables g[1], ..., g[n] is P(n) = (1/(n+1)!)*(d^n/dx^n)(1/G(x)^(n+1))|{x=0}. P(0):=1. (d^k/dx^k)G(x)|{x=0} = k!*g[k], k>=1; G(0)=1.
a(n, k) is the coefficient in P(n) of g[1]^e(k, 1)*g[2]^e(k, 2)*..*g[n]^e(k, n) with the k-th partition of n written as [1^e(k, 1), 2^e(k, 2), ..., n^e(k, n)] in Abramowitz-Stegun order (e(k, j) >= 0; if e(k, j)=0 then j^0 is not recorded).
T(n,k) = (-1)^j*(n+j)!/((n+1)!*Product_{i>=1} s_i!), where (1*s_1 + 2*s_2 + ... = n) is the k-th partition of n and j = s_1 + s_2 ... is the number of parts. - Andrew Howroyd, Feb 01 2022

Name edited by Andrew Howroyd, Feb 02 2022

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

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 *)

A105805 Irregular triangle read by rows: T(n,k) is the Dyson's rank of the k-th partition of n in Abramowitz-Stegun order.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Apr 28 2005

The rank of a partition is the largest part minus the number of parts.
Row lengths give A000041, n >= 1.
Just for n <= 6, row n is antisymmetric due to conjugation of partitions (see links under A105806): a(n, p(n)-(k-1)) = a(n,k), k = 1..floor(p(n)/2). [Comment corrected by Franklin T. Adams-Watters, Jan 17 2006]
First differs from A330368 at a(49) = T(7,5). - Omar E. Pol, Dec 31 2019

			Triangle begins:
  [0];
  [1, -1];
  [2,  0, -2];
  [3,  1,  0, -1, -3];
  [4,  2,  1,  0, -1, -2, -4];
  [5,  3,  2,  1,  1,  0, -1, -1, -2, -3, -5];
  ...
Row 3 for partitions of 3 in the mentioned order: 3,(1,2),1^3 with ranks 2,0,-2.
From _Wolfdieter Lang_, Jul 18 2013: (Start)
Row n = 7 is [6, 4, 3, 2, 2, 1, 0 , 0, 0, -1, -2, -2, -3, -4, -6].
This is also antisymmetric, but by accident, because a(7,7) = 0 for the partition (1,3^2), conjugate to (2^2,3) with a(7,8) = 0, and a(7,9) = 0 for (1^3,4) which is self-conjugate.
Row n=8 (see the link) is no longer antisymmetric. See the _Franklin T. Adams-Watters_ correction above. (End)

A. 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.
Freeman J. Dyson, Problems for solution nr. 4261, Am. Math. Month. 54 (1947) 418.
Wolfdieter Lang, First 16 rows.

Cf. A000041, A036043, A049085, A209616 (sum of positive ranks), A330368 (another version).

# ASPrts is implemented in A119441
A105805 := proc(n,k)
    local pi;
    pi := ASPrts(n)[k] ;
    max(op(pi))-nops(pi) ;
end proc:
for n from 1 do
    for k from 1 to A000041(n) do
        printf("%d,",A105805(n,k)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 17 2013

a(n,k) = A049085(n,k) - A036043(n,k). - Alford Arnold, Aug 02 2010

Name clarified by Omar E. Pol, Dec 31 2019

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}]

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

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 23 2020

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

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

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

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

cp's OEIS Frontend

A115131 Waring numbers for power sums functions in terms of elementary symmetric functions; irregular triangle T(n,k), read by rows, for n >= 1 and 1 <= k <= A000041(n).

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A035206 Number of multisets associated with least integer of each prime signature.

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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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A111785 T(n,k) are coefficients used for power series inversion (sometimes called reversion), n >= 0, k = 1..A000041(n), read by rows.

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

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

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A105805 Irregular triangle read by rows: T(n,k) is the Dyson's rank of the k-th partition of n in Abramowitz-Stegun order.

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