A003040 -id:A003040 - cp's OEIS frontend

A082914 a(n) = n! / A003040(n).

1, 2, 3, 8, 20, 45, 144, 448, 1680, 4725, 17280, 62208, 290304, 1254400, 4465125, 18144000, 72990720, 391910400, 1881169920, 9754214400, 45660160000, 205752960000, 905748480000, 5280992640000, 28326238617600, 162956344320000, 853298675712000, 5309413982208000

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), May 25 2003

nonn

Minimum product of hook lengths of a partition of n. - Eric M. Schmidt, May 07 2013

			For n=4, we can have
abcd, abc and ab (the rest are symmetric).
......d.......cd
The hook products are 4! = 24, 4*2*1*1 = 8 and 3*2*2*1 = 12, so a(4) = 8. - _Jon Perry_

Eric M. Schmidt, Table of n, a(n) for n = 1..80
Eric Weisstein's World of Mathematics, Hook Length Formula

Cf. A003040.

# Maple code from  Emeric Deutsch, May 12 2004 (Start)
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(j)[i])),i=1..numbpart(j))])[1],j=1..30);
# the procedure H gives the hook product for a given partition written with parts in nonincreasing order;
# if in the definition of the procedure a we replace "else 1" by "else x", then the matrix A yields all the hooklengths corresponding to a partition.
# (End)

More terms from Eric M. Schmidt, May 07 2013

A060240 Triangle T(n,k) in which n-th row gives degrees of irreducible representations of symmetric group S_n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 4, 4, 5, 5, 6, 1, 1, 5, 5, 5, 5, 9, 9, 10, 10, 16, 1, 1, 6, 6, 14, 14, 14, 14, 15, 15, 20, 21, 21, 35, 35, 1, 1, 7, 7, 14, 14, 20, 20, 21, 21, 28, 28, 35, 35, 42, 56, 56, 64, 64, 70, 70, 90, 1, 1, 8, 8, 27, 27, 28, 28, 42, 42, 42, 48, 48, 56, 56, 70, 84

Offset: 0

N. J. A. Sloane, Mar 21 2001

Sum_{k>=1} T(n,k)^2 = n!. - R. J. Mathar, May 09 2013
From Emeric Deutsch, Oct 31 2014: (Start)
Number of entries in row n = A000041(n) = number of partitions of n.
Sum of entries in row n = A000085(n).
Largest (= last) entry in row n = A003040(n).
The entries in row n give the number of standard Young tableaux of the Ferrers diagrams of the partitions of n (nondecreasingly). (End)

			Triangle begins:
  1;
  1;
  1, 1;
  1, 1, 2;
  1, 1, 2, 3, 3;
  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.
B. E. Sagan, The Symmetric Group, 2nd ed., Springer, 2001, New York.

Alois P. Heinz, Rows n = 0..36, flattened
J. S. Frame, G. de B. Robinson, and R. M. Thrall, The hook graphs of the symmetric group Canad. J. Math, 6:316-324, 1954. See Theorem 1, p. 318.
Index entries for sequences related to groups

Rows give A003870, A003871, etc. Cf. A060241, A060246, A060247.
Maximal entry in each row gives A003040.
Cf. A000041, A000085, A000142, A060437, A093784, A224653.

CharacterTable(SymmetricGroup(6)); // (say)

h:= proc(l) local n; n:= nops(l); add(i, i=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]}, 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_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, {}]]; T[1] = {1};
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jan 27 2014, after Alois P. Heinz *)

More terms from Vladeta Jovovic, May 20 2003

A067855 Square of the Euclidean length of the vector of Littlewood-Richardson coefficients of Sum_{lambda |- n} s_lambda^2, where s_lambda are the symmetric Schur functions and the sum runs over all partitions lambda of n.

Original entry on oeis.org

1, 2, 8, 26, 94, 326, 1196, 4358, 16248, 60854, 230184, 874878, 3343614, 12825418, 49368388, 190554410, 737328366, 2858974502, 11106267880, 43215101102, 168398785002, 657070401106, 2566847255572, 10038191414610, 39295007540748

Offset: 0

Richard Stanley, Feb 15 2002

Original name: "Squared length of sum of s_lambda^2, where s_lambda is a Schur function and lambda ranges over all partitions of n."
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/2, g(n) = 4. - Seiichi Manyama, Apr 22 2018
The symbol "|-" means "is a partition of", cf. MathWorld link and the Geloun & Ramgoolam paper. The Littlewood-Richardson coefficients allow a product of two Schur functions to be expressed as a linear combination of Schur functions of the corresponding degree. (The Schur functions symmetric in all n variables correspond to Schur polynomials of partitions extended with 0's to length n.) - M. F. Hasler, Jan 19 2020
See A070933 for similar sums of squares of Littlewood-Richardson coefficients. - M. F. Hasler, Jan 20 2020

			For n=3 the s_lambda^2 summed over all partitions of n and decomposed into a sum of Schur functions yields
    s(6) + 2 s(3,3) + 2 s(4,2) + s(5,1) + 2 s(2,2,2) + 2 s(3,2,1) + s(4,1,1)
    + 2 s(2,2,1,1) + s(3,1,1,1) + s(2,1,1,1,1) + s(1,1,1,1,1,1),
  and the sum of the squares of the coefficients {1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1} gives a(3) = 26.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
J. B. Geloun and S. Ramgoolam, Counting Tensor Model Observables and Branched Covers of the 2-Sphere, arXiv preprint arXiv:1307.6490 [hep-th], 2013.
Eric Weisstein's World of Mathematics, Partition.
Wikipedia, Littlewood-Richardson rule, as of Dec 18 2018.
Wikipedia, Schur polynomial, as of Jan 13 2020.

Cf. A001868.
List of partitions: A036037, A080577, A181317, A330370.
Cf. A070933 (Sum_{lambda,mu,nu} (c^{lambda}_{mu,nu})^2, |mu| = |nu| = n).
Cf. A003040 (maximum number of standard tableaux of the Ferrers diagrams of the partitions of n).

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1,
      binomial(n+n, n), add(b(j, 1)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..33);  # Alois P. Heinz, Aug 24 2019

Table[Tr[(Apply[List,
  Sum[Tr[s @@@ LRRule[\[Lambda], \[Lambda]]],
   {\[Lambda], Partitions[n]}]] /. s[] -> 1)^2], {n, 1, 10}];
(* with 'LRRule' defined in http://users.telenet.be/Wouter.Meeussen/ToolBox.nb - Wouter Meeussen, Jan 19 2020 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 1, Binomial[n+n, n],
     Sum[b[j, 1]*b[n - i*j, i-1], {j, 0, n/i}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Jan 02 2022, after Alois P. Heinz *)

A067855_upto(N)=Vec(1/sqrt(prod(i=1,N-1,1-4*'x^i+O('x^N)))) \\ M. F. Hasler, Jan 23 2020

G.f.: 1/sqrt(Product_{i >= 1} (1 - 4*x^i)).
Euler transform of A001868(n)/2. a(n) = Sum_{pi} Product_{m=1..n} binomial(2*p(m), p(m)), where pi runs through all nonnegative solutions of p(1) + 2*p(2) + ... + n*p(n)=n. - Vladeta Jovovic, Mar 25 2006
a(n) ~ 2^(2*n) / sqrt(c*Pi*n), where c = QPochhammer[1/4] = 0.688537537120339... - Vaclav Kotesovec, Apr 22 2018
By definition, a(n) = Sum_{mu |- 2n} c_mu^2 where Sum_{lambda |- n} s_lambda^2 = Sum_{mu |- 2n} c_mu s_mu, where s_lambda are the Schur polynomials (symmetric in 2n variables) and the sums run over all partitions of n resp. 2n. - M. F. Hasler, Jan 19 2020

More terms from Vladeta Jovovic, Mar 25 2006

Name edited by M. F. Hasler following observations by Wouter Meeussen, Jan 17 2020

A066051 Maximal degree of an irreducible representation of the group of n X n signed permutation matrices.

Original entry on oeis.org

1, 2, 3, 8, 20, 80, 210, 672, 2688, 10080, 44352, 236544, 960960, 4324320, 20270250, 104247000, 522762240, 3024552960, 15713497800, 108973522944, 625746401280, 3824005785600, 24049411386000, 160329409240000, 858907549500000, 5226869622374400

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Dec 29 2001

This group is also the automorphism group of the n-dimensional hypercube and the wreath product of the cyclic group C_2 and the symmetric group S_n.
The number of irreducible representations is given in A000712; the order of the group in A000165.
The group is also the Weyl group of type B_n. - Eric M. Schmidt, Sep 21 2013

Roger W. Carter, Finite Groups of Lie Type: Conjugacy Classes And Complex Characters. Wiley, 1985.

Eric M. Schmidt, Table of n, a(n) for n = 1..40

Cf. A000712, A000165, A003040.

to produce a(8): c := CyclicGroup(2); s := SymmetricGroup(8); w := WreathProduct(c,s); Display(CharacterTable(w));

def A066051(n) : return factorial(n) // min(prod(A.hooks()) * prod(B.hooks()) for (A,B) in PartitionTuples(2,n)) # Eric M. Schmidt, Sep 21 2013

More terms from Eric M. Schmidt, Sep 21 2013

A117500 Triangle read by rows in which row n gives the partition of n associated with highest degree representation of symmetric group S_n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Apr 28 2006

Note that a partition and its conjugate give the same degree representation of the symmetric group. We take the lexicographically earlier of the two.

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

J. McKay, The largest degrees of irreducible characters of the symmetric group. Math. Comp. 30 (1976), no. 135, 624-631. (Gives first 75 rows on pp. 627-629.)
J. McKay, Page 1 of 5 pages of tables from Math. Comp. paper
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

See A003040 for much more information. Cf. A060240.

If p_1 >= p_2 >= ... >= p_k is the partition of n, the degree of the representation (given in A003040) is n! * Product_{i

A061220 Least entry in character table of the symmetric group S_n.

Original entry on oeis.org

1, -1, -1, -1, -2, -3, -6, -16, -36, -91, -224, -768, -2420, -7854, -22815, -73008, -292864, -1223040, -5002998, -17592960, -67184000, -279734796, -1183614120, -5844883968, -29448258840, -124619677182, -573333075000, -2764864302200, -13664438287500

Offset: 1

Ola Veshta (olaveshta(AT)my-deja.com), May 30 2001

The maximal value in the character table is the maximal degree of an irreducible representation of S_n and this is in sequence A003040.

			a(3) = -1 because the character table of S_3 is / 1 1 1 / 1 1 -1 / 2 -1 0 /.

Eric M. Schmidt, Table of n, a(n) for n = 1..35

Cf. A003040, A006907.

A061220 := n -> Minimum(List(Irr(CharacterTable("Symmetric", n)), Minimum)); # Eric M. Schmidt, Feb 18 2013

seq(min(map(op,[entries(combinat:-character(n))])),n=1..23); # Robert Israel, Mar 31 2016

a[n_] := With[{S = "S" <> ToString[n]}, FiniteGroupData[S, "CharacterTable"] // Flatten // Min]; Array[a, 10] (* Jean-François Alcover, Mar 31 2016 *)

Corrected and extended by Vladeta Jovovic, May 20 2003

More terms from Eric M. Schmidt, Feb 18 2013

A258124 Triangle read by rows: T(n,k) is the number of partitions of n having k standard tableaux in their Ferrers diagrams (n>=1, k>=1).

Original entry on oeis.org

1, 2, 2, 1, 2, 1, 2, 2, 0, 0, 2, 2, 1, 2, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Emeric Deutsch, Oct 03 2015

Sum of entries in row n is A000041(n) (the number of partitions of n).
Number of entries in row n is A003040(n).
Sum_{k>=1} k*T(n,k) = A000085(n) = number of standard tableaux of size n.
In the Maple program (too slow and too complex) (i) pp(n) yields the prime number preceding n and 1 if n=2; (ii) B(n) yields the partition corresponding to the Heinz number n; (iii) a(n) yields the number of standard tableaux of the Ferrers diagram of the partition B(n); (iv) Q(n) yields the generating polynomial of the partitions of n with respect to the number of standard tableaux of their Ferrers diagrams. For example, Q(4) = 2x + x^2 + 2x^3; indeed, the partitions [4], [1,1,1,1] have each 1 standard tableau, the partition [2,2] has 2 standard tableaux, and the partitions [3,1], [2,1,1] each have 3 standard tableaux.
The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 1, 4] we get 2*2*2*7 = 56.

			T(6,5) = 4 because there are 4 partitions of 6 having 5 standard tableaux in their Ferrers diagrams: [5,1], [3,3], [2,2,2], and [2,1,1,1,1].
Triangle starts:
1;
2;
2,1;
2,1,2;
2,0,0,2,2,1;
2,0,0,0,4,0,0,0,2,2,0,0,0,0,0,1;

Alois P. Heinz, Rows n = 1..14, flattened
FindStat - Combinatorial Statistic Finder, The number of standard Young tableaux of the partition

Cf. A215366, A000041, A003040, A000085.

with(numtheory): pp := proc (n) if n = 2 then 1 else prevprime(n) end if end proc: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: a := proc (n) local pp, FS: pp := proc (n) if n = 1 then 1 elif n = 2 then 1 else prevprime(n) end if end proc: FS := factorset(n): if n = 1 then 1 else add(a(n*pp(FS[j])/FS[j]), j = 1 .. nops(FS)) end if end proc; Q := proc (n) local R, i: R := 0: for i from ithprime(n) to 2^n do if sum(B(i)[j], j = 1 .. nops(B(i))) = n then R := R+x^a(i) else end if end do: sort(R) end proc: T := proc (n, k) options operator, arrow: coeff(Q(n), x, k) end proc: seq(seq(T(n, k), k = 1 .. degree(Q(n))), n = 1 .. 8); # yields sequence in triangular form

A060955 Highest degree of an irreducible representation of the alternating group A_n of degree n.

Original entry on oeis.org

1, 1, 1, 3, 5, 10, 35, 70, 216, 567, 2310, 5775, 21450, 69498, 243243, 1153152, 3620864, 16336320, 64664600, 249420600, 997682400, 5462865408, 21422145536

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001

nonn

Cf. A003040, A003861-A003868.

list:= []; for i in [1..23] do "i = ", i; g:= Alt(i); ct:= CharacterTable(g); "ct =", ct[ #ct][1]; Append(~list, ct[ #ct][1]); end for; // Colva Roney-Dougal (colva(AT)mcs.st-and.ac.uk), Mar 30 2007

More terms from Colva Roney-Dougal (colva(AT)mcs.st-and.ac.uk), Mar 30 2007

Showing 1-8 of 8 results.

cp's OEIS Frontend

A082914 a(n) = n! / A003040(n).

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A060240 Triangle T(n,k) in which n-th row gives degrees of irreducible representations of symmetric group S_n.

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A067855 Square of the Euclidean length of the vector of Littlewood-Richardson coefficients of Sum_{lambda |- n} s_lambda^2, where s_lambda are the symmetric Schur functions and the sum runs over all partitions lambda of n.

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A066051 Maximal degree of an irreducible representation of the group of n X n signed permutation matrices.

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A117500 Triangle read by rows in which row n gives the partition of n associated with highest degree representation of symmetric group S_n.

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A061220 Least entry in character table of the symmetric group S_n.

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A258124 Triangle read by rows: T(n,k) is the number of partitions of n having k standard tableaux in their Ferrers diagrams (n>=1, k>=1).

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