OFFSET
1,29
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.
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.
FORMULA
EXAMPLE
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)
PROG
(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<n-=numbpart(k))||return(c(partitions(k)[n+numbpart(k)])))
CROSSREFS
KEYWORD
nonn,more,tabf
AUTHOR
M. F. Hasler, Jan 21 2020
STATUS
approved