A330985 Irregular table read by rows in which row n gives the Littlewood-Richardson coefficients for the square of the symmetric Schur function corresponding to the n-th partition listed in A036036 (colexicographic order).

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

Offset: 1

Author

M. F. Hasler, Jan 21 2020

Keywords

• nonn
• more
• tabf

Comments

Not only the rows but also the coefficients in each row are listed in one-to-one correspondence with the partitions as listed in the corresponding row of A036036.

The length of row n in this table equals A000041(2|lambda|), the number of partitions of 2|lambda|, where |lambda| is the sum of parts of the n-th partition listed in A036036. There are A000041(k) rows of length A000041(2k), k >= 1.

The graded colexicographic order is also known as "Abramovitz-Stegun" or better Hindenburg order, cf. Luschny link. (This is the lexicographic order of the partitions padded with '0's to length |lambda| and with parts in increasing order, see column "Ref Colex" on the OEIS Wiki page.)

To each partition lambda is associated a Schur polynomial s_lambda through Jacobi's bialternant formula. The Littlewood-Richardson coefficients are the structure constants in the ring of symmetric functions w.r.t. the basis of Schur functions, i.e., they are the coefficients of products s_mu*s_nu written as linear combinations of the Schur functions s_lambda of degree |lambda| = |mu| + |nu|. (To get this well-defined in terms of symmetric functions, we must consider the polynomials s_mu, s_nu also in |lambda| variables.) This table considers the diagonal of this multiplication table, corresponding to squares of Schur polynomial functions.

Sequence A067855 gives the sum of squares of the coefficients of Sum_{mu |- n} s_mu^2. This corresponds to taking the sum, as vectors, of rows of equal length (equivalent to equal |mu|), and then taking the Euclidean norm squared. For example, for mu |- 2 <=> |mu| = 2, take the sum of rows 2 and 3, to get (1, 1, 2, 1, 1), with sum of squares equal to 8 = A067855(2).

It is known that L-R coefficients for products of "rectangular" partitions contain only 0's and 1's (Okada 1998), therefore rows 5, 8, 10, ... are the first rows that may have terms > 1.

Examples

			The 4th partition listed in A036036 is (1,2); the Schur function (s[1,2])^2 is equal to 0*s[6] + 0*s[1,5] + 1*s[2,4] + 1*s[3,3] + 1*s[1,1,4] + 2*s[1,2,3] + 1*s[2,2,2] + 1*s[1,1,1,3] + 1*s[1,1,2,2] + 0*s[1,1,1,1,2] + 0*s[1,1,1,1,1,1], therefore the 4th row is (0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0).
The table starts:
   n | partition mu | 2|mu| | coefficients of (s_mu)^2
  ---+--------------+-------+---------- ----------------
   1 |     (1)      |   2   | (1, 1)
   2 |     (2)      |   4   | (1, 1, 1, 0, 0)
   3 |    (1,1)     |   4   | (0, 0, 1, 1, 1)
   4 |     (3)      |   6   | (1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0)
   5 |    (1,2)     |   6   | (0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0)
   6 |   (1,1,1)    |   6   | (0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1)
		

Links

• Peter Luschny, Counting with partitions.
• OEIS Wiki, Orderings of partitions (a comparison).
• Soichi Okada, Applications of minor summation formulas to rectangular-shaped representations of classical groups, Journal of Algebra, vol. 205, no 2, 1998, pp. 337-367. DOI: 10.1006/jabr.1997.7408.
• 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 (square of the L2-norm of the vector sum of rows of equal length).

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} \\ Initialize the index {leading monomial => partition}
/* The following computes the row corresponding to partition p, but not very efficiently: it requires lots of memory for |mu| >= 4 (<=> |lambda| >= 8). */
c(p, n=vecsum(Vec(p))*2, B=Schur_index(n))={my(S=s(vecsort(Vec(p,-n)))^2, C=Map()); while(S, my(c); 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)]} \\ If debug>0 (\g1), prints the s_lambda when found in s_p^2.
A330985=concat([c([1]),c([2]),c([1,1]),c([3]),c([2,1]),c([1,1,1])])
A330985_row(n)=for(k=1,oo,(0
				

Formula

s_mu^2 = Sum_{k=1..A000041(2|mu|)} T(n,k)*s_{p(k,2|mu|)}, where mu is the n-th partition listed in A036036, p(k,2|mu|) is the k-th partition in row 2|mu| of A036036, and s_mu, s_p are the Schur functions (or polynomials in 2|mu| variables) associated to the partitions mu resp. p.