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Hopf algebras of heap ordered trees and permutations. (English) Zbl 1161.16029

Summary: A standard heap ordered tree with \(n+1\) nodes is a finite rooted tree in which all the nodes except the root are labeled with the natural numbers between \(1\) and \(n\), and that satisfies the property that the labels of the children of a node are all larger than the label of the node. Denote the set of standard heap ordered trees with \(n+1\) nodes by \(\mathcal T_n\). Let \(k\mathcal T=\bigoplus_{n\geq 0}k\mathcal T_n\). It is known that there are Hopf algebra structures on \(k\mathcal T\). Let \(\mathfrak S_n\) denote the symmetric group on \(n\) symbols. Let \(k\mathfrak S=\bigoplus_{n\geq 0}k\mathfrak S_n\). We give a bialgebra structure on \(k\mathfrak S\), and show that there is a natural bialgebra isomorphism from \(k\mathcal T\) to \(k\mathfrak S\).

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
05C05 Trees
16S34 Group rings
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Online Encyclopedia of Integer Sequences:

Number of doubly labeled heap-ordered trees.

References:

[1] DOI: 10.1007/s10801-005-4628-y · Zbl 1094.16024 · doi:10.1007/s10801-005-4628-y
[2] DOI: 10.1016/0021-8693(89)90328-1 · Zbl 0717.16029 · doi:10.1016/0021-8693(89)90328-1
[3] DOI: 10.1006/jabr.1995.1336 · Zbl 0838.05100 · doi:10.1006/jabr.1995.1336
[4] DOI: 10.1016/0021-8693(76)90182-4 · Zbl 0355.20007 · doi:10.1016/0021-8693(76)90182-4
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