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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2605.27828 |
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| _version_ | 1866910264258985984 |
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| author | Amdeberhan, Tewodros Shareshian, John Stanley, Richard P. |
| author_facet | Amdeberhan, Tewodros Shareshian, John Stanley, Richard P. |
| contents | A \emph{sprout sequence} is a sequence
$\frakr=(R_0=1,R_1,R_2,\dots)$ of symmetric functions in the
variables $\bmx=(x_1,x_2,\dots)$ over a field $K$ generated from a
power series $F(t)=1+a_1t+a_2t^2+\cdots$ by the rule $\sum_{n\geq
0}R_nt^n = \prod_{i\geq 1} F(x_it)$. The power series $F(t)$ is
called the \emph{seed} of $\frakr$. This concept originated in the
work of Littlewood and Richardson (though not with the name ``sprout
sequence''), and numerous examples of sprout sequences have appeared
in the literature. They are related to chromatic Tutte polynomials
of complete graphs and complete hypergraphs, binomial posets, upper
homogeneous (upho) posets, topological genera, etc.
We first develop the basic theory of sprout sequences and then look
at the special case $F(t)=\sec(\sqrt{t})$. We give five
characterizations of sprout sequences and consider the expansion
of sprout symmetric functions in terms of well-known symmetric
function bases. The Schur positivity, elementary symmetric function
positivity, and complete homogeneous symmetric function positivity
of $R_n$ for all $n$ are completely characterized using the
Edrei-Thoma theorem from the theory of total positivity.
The seed $F(t)=\sec(\sqrt{t})$ is especially interesting. The
expansion of $R_n$ in the power sum or monomial basis is related to
alternating permutations. The Schur function expansion is related to
standard Young skew tableaux. The expansion in terms of the complete
symmetric functions has nonnegative integer coefficients, but we
don't know a combinatorial interpretation. Finally we give a formula
for $R_n$ as a sum of chromatic symmetric functions of interval
orders. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27828 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sprout Symmetric Functions: Part 1 Amdeberhan, Tewodros Shareshian, John Stanley, Richard P. Combinatorics A \emph{sprout sequence} is a sequence $\frakr=(R_0=1,R_1,R_2,\dots)$ of symmetric functions in the variables $\bmx=(x_1,x_2,\dots)$ over a field $K$ generated from a power series $F(t)=1+a_1t+a_2t^2+\cdots$ by the rule $\sum_{n\geq 0}R_nt^n = \prod_{i\geq 1} F(x_it)$. The power series $F(t)$ is called the \emph{seed} of $\frakr$. This concept originated in the work of Littlewood and Richardson (though not with the name ``sprout sequence''), and numerous examples of sprout sequences have appeared in the literature. They are related to chromatic Tutte polynomials of complete graphs and complete hypergraphs, binomial posets, upper homogeneous (upho) posets, topological genera, etc. We first develop the basic theory of sprout sequences and then look at the special case $F(t)=\sec(\sqrt{t})$. We give five characterizations of sprout sequences and consider the expansion of sprout symmetric functions in terms of well-known symmetric function bases. The Schur positivity, elementary symmetric function positivity, and complete homogeneous symmetric function positivity of $R_n$ for all $n$ are completely characterized using the Edrei-Thoma theorem from the theory of total positivity. The seed $F(t)=\sec(\sqrt{t})$ is especially interesting. The expansion of $R_n$ in the power sum or monomial basis is related to alternating permutations. The Schur function expansion is related to standard Young skew tableaux. The expansion in terms of the complete symmetric functions has nonnegative integer coefficients, but we don't know a combinatorial interpretation. Finally we give a formula for $R_n$ as a sum of chromatic symmetric functions of interval orders. |
| title | Sprout Symmetric Functions: Part 1 |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.27828 |