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a(n,-1,k) is conjectured to also be the count of monomials (or terms) in the Schur polynomials of k variables and degree n, summed over all partitions of n in at most k parts (zero-padded to length k).

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a(3)=85 since the Schur polynomial of 5 variables and degree 4 starts of off as x[1]*x[2]*x[3]*x[4] + x[1]*x[2]*x[3]*x[5] + ... + x[4]*x[5]^3 + x[5]^4. The exponents collect to the padded partitions of 4 as 5*p(1) + 40*p(2) + 30*p(3) + 150*p(4) + 50*p(5) where p(1) is the lexicographically first padded partition of 4: {4,0,0,0}, a coded form of monomials x[i]^4, and p(5) stands for {1,1,1,1}, coding x[i]x[j]x[k]x[l] with all indices different.

proposed

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proposed

a(n3)=85 since the Schur polynomial of 5 variables and degree 4 starts of as x[1]*x[2]*x[3]*x[4] + x[1]*x[2]*x[3]*x[5] + ... + x[4]*x[5]^3 + x[5]^4. The exponents collect to the padded partitions of 4 as 5*p(1) + 40*p(2) + 30*p(3) + 150*p(4) + 50*p(5) where p(1) is the lexicographically first padded partition of 4: {4,0,0,0}, a coded form of monomials x[i]^4, and p(5) stands for {1,1,1,1}, coding x[i]x[j]x[k]x[l] with all indices different.

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Wikipedia, <a href="httphttps://en.wikipedia.org/wiki/Schur_polynomial">Schur Polynomial</a>

nonn,easy

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a(n)=85 since the Schur polynomial of 5 variables and degree 4 starts of as x[1]*x[2]*x[3]*x[4] + x[1]*x[2]*x[3]*x[5] + ... + x[4]*x[5]^3 + x[5]^4. The exponents collect to the padded partitions of 4 as 5*p(1) + 40*p(2) + 30*p(3) + 150*p(4) + 50*p(5) where p(1) is the lexicographically- first padded partition of 4: {4,0,0,0}, a coded form of monomials x[i]^4, and p(5) stands for {1,1,1,1}, coding x[i]x[j]x[k]x[l] with all indices different.

approved

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