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A330986 Irregular table read by rows in which the rows list the Littlewood-Richardson coefficients for products of Schur functions s_mu * s_nu, for partitions mu >= nu in the order they are listed in A036036 (colexicographic order).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0
Offset: 1

Views

Author

M. F. Hasler, Jan 23 2020

Keywords

Comments

To each partition lambda is associated a Schur polynomial s_lambda through Jacobi's bialternant formula. To get the symmetric function corresponding to a product s_mu * s_nu, one must consider both polynomials in |mu|+|nu| variables, as obtained by Jacobi's formula when mu and nu are padded with parts 0 to length |mu|+|nu|. Here |mu| is the sum of parts of mu.
The rows of this table list the Littlewood-Richardson coefficients, structure constants in the ring of symmetric functions w.r.t. the basis of Schur functions, which give a product s_mu * s_nu as linear combination of the s_lambda with the lambda listed in row |mu|+|nu| of A036036.
If mu(n) denotes the n-th partition listed in A036036, the rows of this table correspond s_{mu(i)}*s_{mu(j)} with (i,j) = (1,1), (2,1), (2,2), (3,1), (3,2), (3,3), etc. The sequential number of the row (i,j) is i(i-1)/2 + j, cf. comment from Nov 19 2009 in A000027.
The length of row n = i(i-1)/2 + j equals A000041(|mu(i)| + |mu(j)|), the number of partitions of |mu(i)| + |mu(j)|.
The graded colexicographic order is also known as "Abramovitz-Stegun" or better Hindenburg order, cf. Luschny link. (This is also the lexicographic order of the partitions with parts in increasing order and padded with 0's to length |lambda|, see column "Ref Colex" on the OEIS Wiki page.)
Sequence A330985 gives the subsequence of rows n(n+1)/2 corresponding to the "diagonal" nu = mu (or i = j). See there for the link with sequence A067855.

Examples

			The table starts: (first column = row number, last column =  sequence data.)
   n | (i,j) |  mu   |  nu   ||mu|+|nu|| coefficients of s_mu*s_nu
  ---+-------+-------+-------+---------+--------------------------
   1 | (1,1) |  (1)  |  (1)  |    2    | (1, 1)
   2 | (2,1) |  (2)  |  (1)  |    3    | (1, 1, 0)
   3 | (2,2) |  (2)  |  (2)  |    4    | (1, 1, 1, 0, 0)
   4 | (3,1) | (1,1) |  (1)  |    3    | (0, 1, 1)
   5 | (3,2) | (1,1) |  (2)  |    4    | (0, 1, 0, 1, 0)
   6 | (3,3) | (1,1) | (1,1) |    4    | (0, 0, 1, 1, 1)
   7 | (4,1) |  (3)  |  (1)  |    4    | (1, 1, 0, 0, 0)
   8 | (4,2) |  (3)  |  (2)  |    5    | (1, 1, 1, 0, 0, 0, 0)
   9 | (4,3) |  (3)  | (1,1) |    5    | (0, 1, 0, 1, 0, 0, 0)
  10 | (4,4) |  (3)  |  (3)  |    6    | (1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0)
  11 | (5,1) | (1,2) |  (1)  |    4    | (0, 1, 1, 1, 0)
  12 | (5,2) | (1,2) |  (2)  |    5    | (0, 1, 1, 1, 1, 0, 0)
  13 | (5,3) | (1,2) | (1,1) |    5    | (0, 0, 1, 1, 1, 1, 0)
  14 | (5,4) | (1,2) |  (3)  |    6    | (0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0)
  15 | (5,5) | (1,2) | (1,2) |    6    | (0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0)
Row 1 is (1, 1) since s[1,0] = x1 + x2 squared is 1*s[2,0] + 1*s[1,1], where s[2,0] = x1^2 + x1*x2 + x2^2 and s[1,1] = x1*x2 are the two Schur polynomials associated to the two partitions of 2.
Row 2 is (1, 1, 0) since the product of s[1,0,0] = x1 + x2 + x3 and s[2,0,0]= x1^2 + x2^2 + x3^3 + x1*x2 + x1*x3 + x2*x3 is 1*s[3,0,0] + 1*s[2,1,0] + 0*s[1,1,1], where s[3,0,0] = x1^3 + x1^2*(x2 + x3) + cyclic + x1*x2*x3, s[2,1,0] = x1^2*(x2 + x3) + cyclic + 2*x1*x2*x3 and s[1,1,1] = x1*x2*x3 are the Schur polynomials associated to the three partitions of 3.

Links

Crossrefs

Cf. A000041 (partition numbers), A036036 (partitions in colex order).
Cf. A067855 (sum of squares of coefficients of sum_{mu|-n} s_mu^2).
Cf. A330985 (rows n(n+1)/2 corresponding to nu = mu).

Programs

  • PARI
    s(p,x=eval([Str("'x"i)|i<-[1..#p]]))={my(J(p)=matdet(matrix(#p,#p, i,j, x[i]^p[j]))); J(Vec(p)+[0..#p-1])/J([0..#p-1])} \\ Schur polynomial corresponding to partition p with p(1) <= ... <= p(n) (otherwise the result differs!).
lead(P,m=1)={while(poldegree(P),m*=variable(P)^poldegree(P);P=pollead(P));m} \\ leading monomial of the polynomial P
lcoef(P)={while(poldegree(P),P=pollead(P));P} \\ coeff. of leading monomial
Schur_index(n,B=Map())={forpart(p=n,mapput(B,lead(s(p)),p));B} \\ Compute the index {leading monomial => partition}
Schur_coeff(S, n=#variables(S), B=Schur_index(n))={ my(C=Map(),c,p); while(S, mapput(C, p=mapget(B,lead(S)), c=lcoef(S)); S-=c*s(Vec(p,-n)); if(default(debug), printf("%+d s%d ",c,Vec(p)))); [iferr(mapget(C,p),E,0) | p<-partitions(n)]} \\ Compute coords of S in Schur basis. If debug>0 (\g1), prints the s_lambda when found in s_p^2.
{LR_coeff(mu, nu, n=vecsum(Vec(mu))+vecsum(Vec(nu)))= Schur_coeff(s(vecsort(Vec(mu,-n)))*s(vecsort(Vec(nu,-n))),n)}
P=concat(vector(3,n,partitions(n)))/*first few rows of A036036*/
A=concat(vector(5,i, vector(i,j, LR_coeff(P[i],P[j]))))

Formula

s_mu*s_nu = Sum_{k=1..A000041(|mu|+|nu|)} T(n,k)*s_{p(k,|mu|+|nu|)}, where n = i(i-1)/2 + j if mu and nu are the i-th resp. j-th partition listed in A036036, and p(k,|mu|+|nu|) is the k-th partition in row |mu|+|nu| of A036036.

A332302 Squared length of sum of e_lambda e_lambda', where e_lambda is an elementary symmetric function and lambda ranges over all partitions of n and lambda' is the adjoint (or transpose) of lambda.

Original entry on oeis.org

1, 4, 5, 9, 13, 21, 29, 50, 66, 98, 134, 191, 255, 355, 468, 633, 829, 1117, 1438, 1895, 2432, 3156, 4021, 5163, 6520, 8292, 10406, 13108, 16345, 20438, 25320, 31491, 38797, 47890, 58737, 72105, 87991, 107473, 130577, 158686, 192021, 232328, 279993, 337391, 405112, 486212, 581806, 695763
Offset: 1

Views

Author

Wouter Meeussen, Feb 09 2020

Keywords

Comments

Similar to A067855, but with the elementary symmetric function instead of the Schur function. Note that A067855 describes (s_lambda)^2 which equals the count for (s_lambda . s_lambda'). This is not the case for the other symmetric functions. Squared length of sum of (e_lambda)^2 is simply A000041 (the partition numbers).
The result is identical for the homogenous and the power sum symmetric functions h_lambda and p_lambda since all three can be written as products: e_lambda = Product_{i=1..n} e(lambda_i).

Examples

			For n = 4, we get a(4) = 9 since
e(4)e(1,1,1,1) = e(4,1,1,1,1);
e(3,1)e(2,1,1) = e(3,2,1,1);
e(2,2)e(2,2)   = e(2,2,2,2);
e(2,1,1)e(3,1) = e(3,2,1,1);
e(1,1,1,1)e(4) = e(4,1,1,1,1);
summing to 2 e(4,1,1,1,1) + 2 e(3,2,1,1) + e(2,2,2,2)
with coefficient vector (2,2,1) and length squared 2^2 + 2^2 + 1^2 = 9.

Crossrefs

Cf. A000041, A067855.

Programs

  • Mathematica
    Table[aa = Reverse[Sort[Join[#, TransposePartition[#]]]]&/@ Partitions[n]; (#.#)&@ Map[Last, Tally[aa]], {n, 48}]
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