cp's OEIS Frontend

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).

%I A330985 #28 Feb 21 2023 02:43:23
%S A330985 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,
%T A330985 0,0,0,0,0,0,1,0,1,1,1
%N 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).
%C A330985 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.
%C A330985 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.
%C A330985 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.)
%C A330985 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.
%C A330985 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).
%C A330985 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.
%H A330985 Peter Luschny, <a href="http://www.luschny.de/math/seq/CountingWithPartitions.html">Counting with partitions</a>.
%H A330985 OEIS Wiki, <a href="https://oeis.org/wiki/Orderings of partitions#A_comparison">Orderings of partitions (a comparison)</a>.
%H A330985 Soichi Okada, <a href="https://doi.org/10.1006/jabr.1997.7408">Applications of minor summation formulas to rectangular-shaped representations of classical groups</a>, Journal of Algebra, vol. 205, no 2, 1998, pp. 337-367. DOI: 10.1006/jabr.1997.7408.
%H A330985 Wikipedia, <a href="https://en.wikipedia.org/wiki/Littlewood-Richardson_rule">Littlewood-Richardson rule</a>, as of Dec 18 2018.
%H A330985 Wikipedia, <a href="https://en.wikipedia.org/wiki/Schur_polynomial">Schur polynomial</a>, as of Jan 13 2020.
%F A330985 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.
%e A330985 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).
%e A330985 The table starts:
%e A330985    n | partition mu | 2|mu| | coefficients of (s_mu)^2
%e A330985   ---+--------------+-------+---------- ----------------
%e A330985    1 |     (1)      |   2   | (1, 1)
%e A330985    2 |     (2)      |   4   | (1, 1, 1, 0, 0)
%e A330985    3 |    (1,1)     |   4   | (0, 0, 1, 1, 1)
%e A330985    4 |     (3)      |   6   | (1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0)
%e A330985    5 |    (1,2)     |   6   | (0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0)
%e A330985    6 |   (1,1,1)    |   6   | (0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1)
%o A330985 (PARI)
%o A330985 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!).
%o A330985 lead(P,m=1)={while(poldegree(P),m*=variable(P)^poldegree(P);P=pollead(P));m} \\ leading monomial of the polynomial P
%o A330985 lcoef(P)={while(poldegree(P),P=pollead(P));P} \\ coeff. of leading monomial
%o A330985 Schur_index(n,B=Map())={forpart(p=n,mapput(B,lead(s(p)),p));B} \\ Initialize the index {leading monomial => partition}
%o A330985 /* The following computes the row corresponding to partition p, but not very efficiently: it requires lots of memory for |mu| >= 4 (<=> |lambda| >= 8). */
%o A330985 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.
%o A330985 A330985=concat([c([1]),c([2]),c([1,1]),c([3]),c([2,1]),c([1,1,1])])
%o A330985 A330985_row(n)=for(k=1,oo,(0<n-=numbpart(k))||return(c(partitions(k)[n+numbpart(k)])))
%Y A330985 Cf. A000041 (partition numbers), A036036 (partitions in colex order).
%Y A330985 Cf. A067855 (square of the L2-norm of the vector sum of rows of equal length).
%K A330985 nonn,more,tabf
%O A330985 1,29
%A A330985 _M. F. Hasler_, Jan 21 2020