A060240 -id:A060240 - cp's OEIS frontend

A153734 Triangle T(n,k): T(n,k) gives the A153452(m_k) such that A056239(m_k) = n, [1<=k<=A000041(n)], sorted by m_k, read by rows. Sequence A060240 is this sequence's permutation.

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 4, 5, 5, 6, 4, 1, 1, 9, 5, 5, 5, 10, 16, 9, 10, 5, 1, 1, 6, 14, 14, 35, 15, 21, 21, 14, 20, 35, 14, 15, 6, 1, 1, 7, 20, 14, 21, 28, 56, 64, 70, 42, 14, 90, 35, 70, 56, 28, 35, 64, 20, 21, 7, 1

Offset: 0

Naohiro Nomoto, Dec 31 2008

Lengths of rows are 1, 1, 2, 3, 5, 7, 11, 15, 22, 30,.... (A000041). Row sums give A000085.

			For n=4, A056239(7) = A056239(9) = A056239(10) = A056239(12) = A056239(16) = 4. Hence T(4,k) = A153452(m_k) = (1,2,3,3,1), where 1<=k<=5, m_k = 7,9,10,12,16.
Triangle T(n,k) begins:
  1;
  1;
  1, 1;
  1, 2, 1;
  1, 2, 3, 3, 1;
  1, 4, 5, 5, 6,  4,  1;
  1, 9, 5, 5, 5, 10, 16, 9, 10, 5, 1;
  ...

Alois P. Heinz, Rows n = 0..26, flattened

Cf. A067924, A215366.

with(numtheory):
g:= proc(n) option remember; `if`(n=1, 1,
      add(g(n/q*`if`(q=2, 1, prevprime(q))), q=factorset(n)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n],
       [seq(map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])
    end:
T:= n-> map(g, sort(b(n, n)))[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 09 2012

g[n_] := g[n] = If[n == 1, 1, Sum[g[n/q*If[q == 2, 1, NextPrime[q, -1]]], {q, FactorInteger[n][[All, 1]]}]];
b[n_, i_] := b[n, i] = If[n == 0 || i < 2, {2^n}, Flatten[Table[Map[ #*Prime[i]^j&, b[n - i*j, i - 1]], {j, 0, n/i}]]];
T[n_] := g /@ Sort[b[n, n]];
T /@ Range[0, 10] // Flatten (* Jean-François Alcover, Feb 16 2021, after Alois P. Heinz *)

A067924 Triangle read by rows in which the n-th row gives degrees of irreducible representations of symmetric group S_n (cf. A060240) but now rows are sorted as indicated in A059797 with p(n) terms on each row, where p(n) = A000041(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, 1, 4, 6, 4, 1, 5, 5, 1, 5, 10, 10, 5, 1, 9, 16, 9, 5, 5, 1, 6, 15, 20, 15, 6, 1, 14, 35, 35, 14, 14, 21, 21, 14, 1, 7, 21, 35, 35, 21, 7, 1, 20, 64, 90, 64, 20, 28, 70, 56, 56, 70, 28, 14, 42, 14

Offset: 1

Alford Arnold, Mar 04 2002

These are Betti numbers [Dochtermann]. - N. J. A. Sloane, Dec 02 2015
Number of terms in row n is A000041(n).
Row sums generate sequence A000085: 1, 2, 4, 10, 26, 76, ...
Sum of squares generates A000142; e.g., -1*1 + 4*4 + 6*6 + 4*4 + 1*1 + 5*5 +5*5 = 5! = 120.

			A059797 begins 2, 5, 5, 9, 16, 9, so row six of this sequence begins 1, 5, 10, 10, 5, 1, 9, 16, 9, ...
Triangle begins:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  3,  1,  2;
  1, 4,  6,  4,  1,  5, 5;
  1, 5, 10, 10,  5,  1, 9, 16,  9,  5,  5;
  1, 6, 15, 20, 15,  6, 1, 14, 35, 35, 14, 14, 21, 21, 14;
  1, 7, 21, 35, 35, 21, 7,  1, 20, 64, 90, 64, 20, 28, 70, 56, 56, 70, 28, 14, 42, 14;

A. Dochtermann, Face rings of cycles, associahedra, and standard Young tableaux, arXiv preprint arXiv:1503.06243 [math.CO], 2015-2016.

Cf. A000041, A000085, A000142, A059797, A060240.

A117506 Irregular triangle read by rows: dimensions of the irreducible representations of the symmetric group S_n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 3, 1, 1, 4, 5, 6, 5, 4, 1, 1, 5, 9, 5, 10, 16, 5, 10, 9, 5, 1, 1, 6, 14, 14, 15, 35, 21, 21, 20, 35, 14, 15, 14, 6, 1, 1, 7, 20, 28, 14, 21, 64, 70, 56, 42, 35, 90, 56, 70, 14, 35, 64, 28, 21, 20, 7, 1

Offset: 0

Wolfdieter Lang, Apr 13 2006

The n-th row has partition(n) = A000041(n) entries.
Also the numbers of standard Young tableaux for Young diagrams (or partitions).
Also "generalized" Catalan numbers. For a partition of n, n=(n_1+...+n_d), this is the number of integral lattice paths from (0,...,0) to (n_1,...,n_d) such that for any point p=(p_1,...p_d) on such a path p_i is never less than p_j whenever iGraham H. Hawkes, Jul 05 2013
The irreducible representations of S_n correspond to Young diagrams or partitions.
Partitions of n are ordered according to Abramowitz-Stegun (A-St) (see the reference, pp. 831-2). In contrast to A-St, a partition has nondecreasing parts (reverse notation of A-St).
The dimension of a representation of S_n corresponding to a Young diagram or partition is a(n,k) for the k-th partition of n in this A-St order.
One could call these numbers a(n,k) M_4 (similar to M_0, M_1, M_2, M_3 given in A111786, A036038, A036039, A036040, respectively).
From Wolfdieter Lang, Oct 09 2015: (Start)
The first formula given below appears in A. Young, Q.S.A. III, PLMS 28 (1928) 255-292 (third paper on "On Quantitative Substitutional Analysis"), Theorem II on p. 260, and he calls it f; see the collected papers (CP) reference, p. 357. Note the shorthand notation for the products; see Q.S.A. II, PLMS 34 (1902) 361-397, p. 366, CP, p. 97, for the explicit one.
This formula also can be found in the Glass-Ng link, Theorem 1, p. 702, using the Vandermonde determinant in the numerator and re-indexing the denominator.
The product of the hook length numbers, called H(n, k) in this formula below, is found in A263003(n, k).
The squared row entries sum to n!. See A. Young, Q.S.A. II (see above), pp. 367-368, CP pp. 98-99. Also Q.S.A. III, p. 265, CP p. 362.
(End)

			[1];
[1];
[1, 1];
[1, 2, 1];
[1, 3, 2, 3, 1];
[1, 4, 5, 6, 5, 4, 1];
[1, 5, 9, 5, 10, 16, 5, 10, 9, 5, 1];...
a(4,4)=3 because the 4th partition of n=4 in A-St order is [2,1,1],
and H(4,4)=(4!*2!*1!)/Vandermonde([4,2,1]) = (4!*2)/6 =4*2, hence
4!/H(4,4) = 3.
a(4,4)=3 because the hook lengths of the Young diagram of [2,1,1] are [4, 1; 2; 1], hence 4!/(4*1*2*1) = 3.
The sum of the squared entries of each row gives n!: n = 5: 2*(1^1 + 4^2 + 5^2) + 6^2 = 120 = 5!. - _Wolfdieter Lang_, Oct 09 2015

G. de B. Robinson (ed.), The Collected Papers of Alfred Young 1873-1940, University of Toronto Press, 1977.
G. B. Wybourne, Symmetry principles and atomic spectroscopy, Wiley, New York, 1970, p. 9.

Alois P. Heinz, Rows n = 0..30, 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].
A. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, pp. 831-2.
Kenneth Glass and Chi-Keung Ng, A Simple Proof of the Hook Length Formula, Am. Math. Monthly 111 (2004) 700 - 704.
Graham H. Hawkes, An Elementary Proof of a Formula for SYT, arXiv preprint arXiv:1310.5919 [math.CO], 2013-2014.
Wolfdieter Lang, First 15 rows.
Eric Weisstein's World of Mathematics, Hook length formula.
Doron Zeilberger, Andre's Reflection Proof Generalized to the Many-Candidate Ballot Problem, Discrete Mathematics 44 (1983) 325-326.
Index entries for sequences related to groups

Cf. A000041, A000085 (row sums), A060240 (rows sorted), A263003.

Cf. A067924.

h:= l-> (n-> mul(mul(1+l[i]-j+add(`if`(l[k]>=j, 1, 0),
             k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
g:= (n, i, l)-> `if`(n=0 or i=1, [h([l[], 1$n])],
    [g(n, i-1, l)[], g(n-i, min(n-i, i), [l[], i])[]]):
T:= n-> map(x-> n!/x, g(n$2, []))[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 05 2015

h[l_List] := Function[n, Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]][Length[l]]; g[n_, i_, l_List] := If[n==0 || i==1, Join[{h[Join[l, Array[1&, n]]]}], If[i<1, {}, Join[{g[n, i-1, l]}, If[i>n, {}, g[n-i, i, Join[l, {i}]]]]]] // Flatten; T[n_] := n!/ g[n, n, {}]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 19 2015, after Alois P. Heinz *)

a(n,k) = n!/H(n,k) with H(n,k):= Product_{i=1..m(n,k)} (x_i)!/Det(x_i^(m(n,k)-j)) with the Vandermonde determinant for the variables x_i:=lambda(n,k)_i + m(n,k)-i, i,j=1..m(n,k) if m(n,k) is the number of parts of the k-th partition of n, called lambda(n,k), in the A-St order (see above). Lambda(n,k)_i denotes the i-th part of the partition lambda(n,k), sorted in decreasing order (this is the reverse of the A-St notation).

a(n,k) = n!/Product_{j=1..n}(h(n,k,j) with the hook numbers h(n,k,j) of the Young diagram of the partition lambda(n,k) in the A-St order. See the link for 'hook length formula'.

Row n=0 prepended by Alois P. Heinz, Nov 05 2015

A093784 Triangle T(n,k) read by rows in which n-th row gives the hook products of the partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 6, 6, 8, 8, 12, 24, 24, 20, 24, 24, 30, 30, 120, 120, 45, 72, 72, 80, 80, 144, 144, 144, 144, 720, 720, 144, 144, 240, 240, 252, 336, 336, 360, 360, 360, 360, 840, 840, 5040, 5040, 448, 576, 576, 630, 630, 720, 720, 960, 1152, 1152, 1440, 1440, 1920

Offset: 0

Emeric Deutsch, May 17 2004

Row n consists of the numbers n!/A060240(n,k) written in reverse order.

			Triangle T(n,k) begins:
   1;
   1;
   2,  2;
   3,  6,  6;
   8,  8, 12, 24, 24;
  20, 24, 24, 30, 30, 120, 120;
  45, 72, 72, 80, 80, 144, 144, 144, 144, 720, 720;
  ...

Alois P. Heinz, Rows n = 0..36, flattened

Cf. A060240, A093716, A093764.

H:=proc(pa) local F,j,p,Q,i,col,a,A: F:=proc(x) local i, ct: ct:=0: for i from 1 to nops(x) do if x[i]>1 then ct:=ct+1 else fi od: ct; end: for j from 1 to nops(pa) do p[1][j]:=pa[j] od: Q[1]:=[seq(p[1][j],j=1..nops(pa))]: for i from 2 to pa[1] do for j from 1 to F(Q[i-1]) do p[i][j]:=Q[i-1][j]-1 od: Q[i]:=[seq(p[i][j],j=1..F(Q[i-1]))] od: for i from 1 to pa[1] do col[i]:=[seq(Q[i][j]+nops(Q[i])-j,j=1..nops(Q[i]))] od: a:=proc(i,j) if i<=nops(Q[j]) and j<=pa[1] then Q[j][i]+nops(Q[j])-i else 1 fi end: A:=matrix(nops(pa),pa[1],a): product(product(A[m,n],n=1..pa[1]),m=1..nops(pa)); end: with(combinat): rev:=proc(a) [seq(a[nops(a)+1-i],i=1..nops(a))] end: seq(sort([seq(H(rev(partition(s)[q])),q=1..numbpart(s))]),s=1..9);
# second Maple program:
h:= proc(l) local n; n:= nops(l); mul(mul(1+l[i]-j+
      add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n]), `if`(i<1, 0,
                 seq(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
T:= n-> sort([g(n, n, [])])[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Jan 07 2013

h[l_List] := With[{n = Length[l]},  Product[Product[1+l[[i]]-j+Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[1, 1, {}] = {1}; g[n_, i_, l_List] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]], If[i<1, 0, Flatten @  Table[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; T[n_] := Sort[g[n, n, {}]]; Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Apr 29 2015, after Alois P. Heinz *)

A003040 Highest degree of an irreducible representation of symmetric group S_n of degree n.

Original entry on oeis.org

1, 1, 2, 3, 6, 16, 35, 90, 216, 768, 2310, 7700, 21450, 69498, 292864, 1153152, 4873050, 16336320, 64664600, 249420600, 1118939184, 5462865408, 28542158568, 117487079424, 547591590000, 2474843571200, 12760912164000, 57424104738000, 295284192952320

Offset: 1

N. J. A. Sloane and Richard Stanley

nonn

Highest number of standard tableaux of the Ferrers diagrams of the partitions of n. Example: a(4) = 3 because to the partitions 4, 31, 22, 211, and 1111 there correspond 1, 3, 2, 3, and 1 standard tableaux, respectively. - Emeric Deutsch, Oct 02 2015

			a(5) = 6 because the degrees for S_5 are 1,1,4,4,5,5,6.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
D. E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups. 2nd ed., Oxford University Press, 1950, p. 265.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vasilii Duzhin, Table of n, a(n) for n = 1..153 (terms up to a(80) from Eric M. Schmidt)
S. Comét, Improved methods to calculate the characters of the symmetric group, Math. Comp. 14 (1960) 104-117.
J. McKay, The largest degrees of irreducible characters of the symmetric group. Math. Comp. 30 (1976), no. 135, 624-631. (Gives first 75 terms.)
J. McKay, Page 1 of 5 pages of tables from Math. Comp. paper [reports 29th term incorrectly]
J. McKay, Page 2 of 5 pages of tables from Math. Comp. paper
J. McKay, Page 3 of 5 pages of tables from Math. Comp. paper
J. McKay, Page 4 of 5 pages of tables from Math. Comp. paper
J. McKay, Page 5 of 5 pages of tables from Math. Comp. paper
Igor Pak, Greta Panova, and Damir Yeliussizov, On the largest Kronecker and Littlewood-Richardson coefficients, arXiv:1804.04693 [math.CO], 2018.
R. P. Stanley, Letter to N. J. A. Sloane, c. 1991

A117500 gives the corresponding partitions of n.

Cf. A003869, A003870, A003871, A003872, A003873, A003874, A003875, A003876, A003877.

h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1 &, n]]], If[i < 1, 0, Flatten@ Table[g[n - i*j, i - 1, Join[l, Array[i&, j]]], {j, 0, n/i}]]];
a[n_] := a[n] = g[n, n, {}] // Max;
Table[Print[n, " ", a[n]]; a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 23 2024, after Alois P. Heinz in A060240 *)

def A003040(n):
    res = 1
    for P in Partitions(n):
        res = max(res, P.dimension())
    return res
# Eric M. Schmidt, May 07 2013

Entry revised and extended by N. J. A. Sloane, Apr 28 2006

a(29) corrected by Eric M. Schmidt, May 07 2013

A003875 Degrees of irreducible representations of symmetric group S_11.

Original entry on oeis.org

1, 1, 10, 10, 44, 44, 45, 45, 110, 110, 120, 120, 132, 132, 165, 165, 210, 210, 231, 231, 252, 330, 330, 385, 385, 462, 462, 550, 550, 594, 594, 660, 660, 693, 693, 825, 825, 924, 924, 990, 990, 990, 990, 1100, 1100, 1155, 1155, 1188, 1232, 1232, 1320, 1320, 1540, 1540, 2310, 2310

Offset: 1

N. J. A. Sloane

All 56 terms of this finite sequence are shown.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].

Row n=11 of A060240.

A003875 := List(Irr(CharacterTable("S11")), chi->chi[1]);; Sort(A003875); # Eric M. Schmidt, Jul 18 2012

t1 := CharacterTable(SymmetricGroup(11)); [Degree(t1[i]) : i in [1 .. #t1]];

h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1 &, n]]], If[i < 1, 0, Flatten@ Table[g[n - i*j, i - 1, Join[l, Array[i&, j]]], {j, 0, n/i}]]];
T[n_] := g[n, n, {}];
Sort[T[11]] (* Jean-François Alcover, Sep 23 2024, after Alois P. Heinz in A060240 *)

A003869 Degrees of irreducible representations of symmetric group S_5.

Original entry on oeis.org

1, 1, 4, 4, 5, 5, 6

Offset: 1

N. J. A. Sloane

All 7 terms of this finite sequence are shown.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].

Row n=5 of A060240.

Cf. A003870, A003871, A003872, A003873.

A003869 := List(Irr(CharacterTable("S5")), chi->chi[1]);; Sort(A003869); # Eric M. Schmidt, Jul 18 2012

Magma
```
// See A003875 for Magma code
```

h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]], If[i < 1, 0, Flatten@ Table[g[n - i*j, i - 1, Join[l, Array[i &, j]]], {j, 0, n/i}]]];
T[n_] := g[n, n, {}];
Sort[T[5]] (* Jean-François Alcover, Sep 22 2024, after Alois P. Heinz in A060240 *)

A003870 Degrees of irreducible representations of symmetric group S_6.

Original entry on oeis.org

1, 1, 5, 5, 5, 5, 9, 9, 10, 10, 16

Offset: 1

N. J. A. Sloane

All 11 terms of this finite sequence are shown.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].

Row n=6 of A060240.

A003870 := List(Irr(CharacterTable("S6")), chi->chi[1]);; Sort(A003870); # Eric M. Schmidt, Jul 18 2012

Magma
```
CharacterTable(SymmetricGroup(6));
```

h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1 &, n]]], If[i < 1, 0, Flatten@ Table[g[n - i*j, i - 1, Join[l, Array[i &, j]]], {j, 0, n/i}]]];
T[n_] := g[n, n, {}];
Sort[T[6]] (* Jean-François Alcover, Sep 22 2024, after Alois P. Heinz in A060240 *)

A027840 Number of subgroups of index n in fundamental group of a certain fiber space.

Original entry on oeis.org

1, 15, 220, 5275, 151086, 6605004, 362069912, 26370058035, 2384037107365, 264380945199210, 35133143655934644, 5515729438742221708, 1009373492449379367974, 212997911074525038601560, 51337590023913924398371080, 14016616814674335739387516003

Offset: 1

N. J. A. Sloane

nonn

Sean A. Irvine, Java program (github).
V. A. Liskovets and A. Mednykh, Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.

Cf. A027838, A027839, A027841, A060240.

Third term corrected from 240 to 220, Aug 15 1999

More terms from Sean A. Irvine, Dec 07 2019

A003871 Degrees of irreducible representations of symmetric group S_7.

Original entry on oeis.org

1, 1, 6, 6, 14, 14, 14, 14, 15, 15, 20, 21, 21, 35, 35

Offset: 1

N. J. A. Sloane

All 15 terms of this finite sequence are shown.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].

Row n=7 of A060240.

A003871 := List(Irr(CharacterTable("S7")), chi->chi[1]);; Sort(A003871); # Eric M. Schmidt, Jul 18 2012

h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1 &, n]]], If[i < 1, 0, Flatten@ Table[g[n - i*j, i - 1, Join[l, Array[i&, j]]], {j, 0, n/i}]]];
T[n_] := g[n, n, {}];
Sort[T[7]] (* Jean-François Alcover, Sep 22 2024, after Alois P. Heinz in A060240 *)

cp's OEIS Frontend

A153734 Triangle T(n,k): T(n,k) gives the A153452(m_k) such that A056239(m_k) = n, [1<=k<=A000041(n)], sorted by m_k, read by rows. Sequence A060240 is this sequence's permutation.

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A067924 Triangle read by rows in which the n-th row gives degrees of irreducible representations of symmetric group S_n (cf. A060240) but now rows are sorted as indicated in A059797 with p(n) terms on each row, where p(n) = A000041(n).

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A117506 Irregular triangle read by rows: dimensions of the irreducible representations of the symmetric group S_n.

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A093784 Triangle T(n,k) read by rows in which n-th row gives the hook products of the partitions of n.

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Maple

Mathematica

A003040 Highest degree of an irreducible representation of symmetric group S_n of degree n.

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Mathematica

Sage

Extensions

A003875 Degrees of irreducible representations of symmetric group S_11.

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GAP

Magma

Mathematica

A003869 Degrees of irreducible representations of symmetric group S_5.

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GAP

Magma

Mathematica