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
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
Programs
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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]))))
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