Return to Question

Recent references on the matter at hand include, a lecture slide The Konvalinka-Amdeberhan conjecture and plethystic inverses and a preprint on Counting tanglegrams with species by I. Gessel; the initial work On the enumeration of tanglegrams and tangled chains by S. Billey et al.

For each prime $p$, let's consider the family of (integral) sequences $$a_p(n):=\sum_{\lambda\vdash n}\frac{1}{z_\lambda}\prod_{i=2}^{\ell(\lambda)}(p\lambda_i+\cdots+p\lambda_{\ell(\lambda)}+1);$$ where the sum runs through all $p$-ary partitions $\lambda$, of $n$, and $\ell(\lambda)$ is the length of the partition and the numbers $z_{\lambda}$ are well-known since the number of permutations in $\mathfrak{S}_n$ with cycle type $\lambda$ is computed by $\frac{n!}{z_{\lambda}}$.

On the other hand, Michael Somos proposed the functional equations $x\cdot A_p(x)^p=\frac{x\cdot A_p(x^p)}{1-p\cdot x\cdot A_p(x^p)}$ and a couple of these are listed on OEIS as A085748 and A091190. However, there are not enough interpretations attached to $A_p(x)$ or its coefficients on OEIS.

After some experimentation, I wish to ask:

QUESTION. Can the following be justified or refuted? $$A_p(x)=\sum_{n\geq0}a_p(n)x^n.$$

Remark. Observe that neither $A_p(x)$ nor $a_p(n)$ have been found to output integers, unless $p$ is a prime.

Recent references on the matter at hand include, a lecture slide The Konvalinka-Amdeberhan conjecture and plethystic inverses and a preprint on Counting tanglegrams with species by I. Gessel; the initial work On the enumeration of tanglegrams and tangled chains by S. Billey et al.

For each prime $p$, let's consider the family of (integral) sequences $$a_p(n):=\sum_{\lambda\vdash n}\frac{1}{z_\lambda}\prod_{i=2}^{\ell(\lambda)}(p\lambda_i+\cdots+p\lambda_{\ell(\lambda)}+1);$$ where the sum runs through all $p$-ary partitions $\lambda$, of $n$, and $\ell(\lambda)$ is the length of the partition and the numbers $z_{\lambda}$ are well-known since the number of permutations in $\mathfrak{S}_n$ with cycle type $\lambda$ is computed by $\frac{n!}{z_{\lambda}}$.

On the other hand, Michael Somos proposed the functional equations $$ x\cdot A_p(x)^p=\frac{x\cdot A_p(x^p)}{1-p\cdot x\cdot A_p(x^p)} $$ and a couple of these are listed on OEIS as A085748 and A091190. However, there are not enough interpretations attached to $A_p(x)$ or its coefficients on OEIS.

After some experimentation, I wish to ask:

QUESTION. Can the following be justified or refuted? $$A_p(x)=\sum_{n\geq0}a_p(n)x^n.$$

Remark. Observe that neither $A_p(x)$ nor $a_p(n)$ have been found to output integers, unless $p$ is a prime.

Recent references on the matter at hand include, a lecture slide The Konvalinka-Amdeberhan conjecture and plethystic inverses and a preprint on Counting tanglegrams with species by I. Gessel; the initial work On the enumeration of tanglegrams and tangled chains by S. Billey et al.

For each prime $p$, let's consider the family of (integral) sequences $$a_p(n):=\sum_{\lambda\vdash n}\prod_{i=2}^{\ell(\lambda)}\frac{p\lambda_i+\cdots+p\lambda_{\ell(\lambda)}+1}{z_{\lambda}};$$ where the sum runs through all $p$-ary partitions $\lambda$, of $n$, and $\ell(\lambda)$ is the length of the partition and the numbers $z_{\lambda}$ are well-known since the number of permutations in $\mathfrak{S}_n$ with cycle type $\lambda$ is computed by $\frac{n!}{z_{\lambda}}$.

On the other hand, Michael Somos proposed the functional equations $x\cdot A_p(x)^p=\frac{x\cdot A_p(x^p)}{1-p\cdot x\cdot A_p(x^p)}$ and a couple of these are listed on OEIS as A085748 and A091190. However, there are not enough interpretations attached to $A_p(x)$ or its coefficients on OEIS.

After some experimentation, I wish to ask:

QUESTION. Can the following be justified or refuted? $$A_p(x)=\sum_{n\geq0}a_p(n)x^n.$$

Remark. Observe that neither $A_p(x)$ nor $a_p(n)$ have been found to output integers, unless $p$ is a prime.

Recent references on the matter at hand include, a lecture slide The Konvalinka-Amdeberhan conjecture and plethystic inverses and a preprint on Counting tanglegrams with species by I. Gessel; the initial work On the enumeration of tanglegrams and tangled chains by S. Billey et al.

For each prime $p$, let's consider the family of (integral) sequences $$a_p(n):=\sum_{\lambda\vdash n}\frac{1}{z_\lambda}\prod_{i=2}^{\ell(\lambda)}(p\lambda_i+\cdots+p\lambda_{\ell(\lambda)}+1);$$ where the sum runs through all $p$-ary partitions $\lambda$, of $n$, and $\ell(\lambda)$ is the length of the partition and the numbers $z_{\lambda}$ are well-known since the number of permutations in $\mathfrak{S}_n$ with cycle type $\lambda$ is computed by $\frac{n!}{z_{\lambda}}$.

On the other hand, Michael Somos proposed the functional equations $x\cdot A_p(x)^p=\frac{x\cdot A_p(x^p)}{1-p\cdot x\cdot A_p(x^p)}$ and a couple of these are listed on OEIS as A085748 and A091190. However, there are not enough interpretations attached to $A_p(x)$ or its coefficients on OEIS.

After some experimentation, I wish to ask:

QUESTION. Can the following be justified or refuted? $$A_p(x)=\sum_{n\geq0}a_p(n)x^n.$$

Remark. Observe that neither $A_p(x)$ nor $a_p(n)$ have been found to output integers, unless $p$ is a prime.