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).

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

Author

M. F. Hasler, Jan 23 2020

Keywords

• nonn
• tabf

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

• P. Luschny, Counting with partitions.
• OEIS Wiki, Orderings of partitions (a comparison).
• Wikipedia, Littlewood-Richardson rule, as of Dec 18 2018.
• Wikipedia, Schur polynomial, as of Jan 13 2020.

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.