cp's OEIS Frontend

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).

%I A330986 #11 Apr 06 2020 19:00:03
%S A330986 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,
%T A330986 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,
%U A330986 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
%N 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).
%C A330986 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.
%C A330986 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.
%C A330986 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.
%C A330986 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)|.
%C A330986 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.)
%C A330986 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.
%H A330986 P. Luschny, <a href="http://www.luschny.de/math/seq/CountingWithPartitions.html">Counting with partitions</a>.
%H A330986 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions#A_comparison">Orderings of partitions (a comparison)</a>.
%H A330986 Wikipedia, <a href="https://en.wikipedia.org/wiki/Littlewood-Richardson_rule">Littlewood-Richardson rule</a>, as of Dec 18 2018.
%H A330986 Wikipedia, <a href="https://en.wikipedia.org/wiki/Schur_polynomial">Schur polynomial</a>, as of Jan 13 2020.
%F A330986 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.
%e A330986 The table starts: (first column = row number, last column =  sequence data.)
%e A330986    n | (i,j) |  mu   |  nu   ||mu|+|nu|| coefficients of s_mu*s_nu
%e A330986   ---+-------+-------+-------+---------+--------------------------
%e A330986    1 | (1,1) |  (1)  |  (1)  |    2    | (1, 1)
%e A330986    2 | (2,1) |  (2)  |  (1)  |    3    | (1, 1, 0)
%e A330986    3 | (2,2) |  (2)  |  (2)  |    4    | (1, 1, 1, 0, 0)
%e A330986    4 | (3,1) | (1,1) |  (1)  |    3    | (0, 1, 1)
%e A330986    5 | (3,2) | (1,1) |  (2)  |    4    | (0, 1, 0, 1, 0)
%e A330986    6 | (3,3) | (1,1) | (1,1) |    4    | (0, 0, 1, 1, 1)
%e A330986    7 | (4,1) |  (3)  |  (1)  |    4    | (1, 1, 0, 0, 0)
%e A330986    8 | (4,2) |  (3)  |  (2)  |    5    | (1, 1, 1, 0, 0, 0, 0)
%e A330986    9 | (4,3) |  (3)  | (1,1) |    5    | (0, 1, 0, 1, 0, 0, 0)
%e A330986   10 | (4,4) |  (3)  |  (3)  |    6    | (1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0)
%e A330986   11 | (5,1) | (1,2) |  (1)  |    4    | (0, 1, 1, 1, 0)
%e A330986   12 | (5,2) | (1,2) |  (2)  |    5    | (0, 1, 1, 1, 1, 0, 0)
%e A330986   13 | (5,3) | (1,2) | (1,1) |    5    | (0, 0, 1, 1, 1, 1, 0)
%e A330986   14 | (5,4) | (1,2) |  (3)  |    6    | (0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0)
%e A330986   15 | (5,5) | (1,2) | (1,2) |    6    | (0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0)
%e A330986 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.
%e A330986 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.
%o A330986 (PARI)
%o A330986 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 A330986 lead(P,m=1)={while(poldegree(P),m*=variable(P)^poldegree(P);P=pollead(P));m} \\ leading monomial of the polynomial P
%o A330986 lcoef(P)={while(poldegree(P),P=pollead(P));P} \\ coeff. of leading monomial
%o A330986 Schur_index(n,B=Map())={forpart(p=n,mapput(B,lead(s(p)),p));B} \\ Compute the index {leading monomial => partition}
%o A330986 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.
%o A330986 {LR_coeff(mu, nu, n=vecsum(Vec(mu))+vecsum(Vec(nu)))= Schur_coeff(s(vecsort(Vec(mu,-n)))*s(vecsort(Vec(nu,-n))),n)}
%o A330986 P=concat(vector(3,n,partitions(n)))/*first few rows of A036036*/
%o A330986 A=concat(vector(5,i, vector(i,j, LR_coeff(P[i],P[j]))))
%Y A330986 Cf. A000041 (partition numbers), A036036 (partitions in colex order).
%Y A330986 Cf. A067855 (sum of squares of coefficients of sum_{mu|-n} s_mu^2).
%Y A330986 Cf. A330985 (rows n(n+1)/2 corresponding to nu = mu).
%K A330986 nonn,tabf
%O A330986 1,89
%A A330986 _M. F. Hasler_, Jan 23 2020