Eulerian insertion operators and an Eulerian form of the Pieri rule
Abstract
We study the operators obtained by inserting copies of a new largest letter into multiset permutations. Let Gr denote the operator which inserts r copies of a new largest letter. After the change of variables δ=y−x, u=x/y, and E=u∂u, we find that Gr=δr/r!E(E+1)⋯(E+r−1). Its generating series acts by a rational substitution, which yields the composition law. Our main result gives a common symmetric-function explanation for the ordinary and major-index operators. For N≥ 0, define ΦN(FN,S)=x|S|+1yN−|S|. We prove that multiplication by the complete homogeneous symmetric function hr becomes the ordinary insertion operator: ΦN+r(hr f)=GrΦN(f), where f∈QSymN. There is a parallel specialization for the major index. A reverse finite principal specialization sends multiplication by hr to an operator Qr, which is a polynomial in the q-shift Θqf(t)=f(qt). Thus the ordinary and major-index operators arise from the same multiplication operator f↦ hr f. Since the functions hr freely generate the ring of symmetric functions, the assignment hr↦ Gr extends to an algebra homomorphism. We determine the kernel of this homomorphism and the image of every homogeneous component. The images of Schur functions satisfy the Littlewood--Richardson multiplication identities, and the one-row case gives an Eulerian form of the Pieri rule.
Related
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BibTeX
@misc{ma2026eulerian,
title = {{Eulerian insertion operators and an Eulerian form of the Pieri rule}},
author = {Shi-Mei Ma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03881},
eprint = {2609.03881},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03881},
}
RIS
TY - UNPB
TI - Eulerian insertion operators and an Eulerian form of the Pieri rule
AU - Shi-Mei Ma
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03881
UR - https://arxiv.org/abs/2609.03881
AB - We study the operators obtained by inserting copies of a new largest letter into multiset permutations. Let G_r denote the operator which inserts r copies of a new largest letter. After the change of variables δ=y−x, u=x/y, and E=u∂_u, we find that G_r=δ^r/r!E(E+1)⋯(E+r−1). Its generating series acts by a rational substitution, which yields the composition law. Our main result gives a common symmetric-function explanation for the ordinary and major-index operators. For N≥ 0, define Φ_N(F_N,S)=x^|S|+1y^N−|S|. We prove that multiplication by the complete homogeneous symmetric function h_r becomes the ordinary insertion operator: Φ_N+r(h_r f)=G_rΦ_N(f), where f∈QSym_N. There is a parallel specialization for the major index. A reverse finite principal specialization sends multiplication by h_r to an operator Q_r, which is a polynomial in the q-shift Θ_qf(t)=f(qt). Thus the ordinary and major-index operators arise from the same multiplication operator f↦ h_r f. Since the functions h_r freely generate the ring of symmetric functions, the assignment h_r↦ G_r extends to an algebra homomorphism. We determine the kernel of this homomorphism and the image of every homogeneous component. The images of Schur functions satisfy the Littlewood--Richardson multiplication identities, and the one-row case gives an Eulerian form of the Pieri rule.
ER -