Refined Enumeration of $${{\varvec{k}}}$$-plane Trees and $${\varvec{k}}$$-noncrossing Trees
Abstract A k - plane tree is a plane tree whose vertices are assigned labels between 1 and k in such a way that the sum of the labels along any edge is no greater than $$k+1$$ k + 1 . These trees are known to be related to $$(k+1)$$ ( k + 1 ) -ary trees, and they are counted by a generalised version...
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| Published in | Annals of combinatorics Vol. 28; no. 1; pp. 121 - 153 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
01.03.2024
|
| Online Access | Get full text |
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| Summary: | Abstract
A
k
-
plane tree
is a plane tree whose vertices are assigned labels between 1 and
k
in such a way that the sum of the labels along any edge is no greater than
$$k+1$$
k
+
1
. These trees are known to be related to
$$(k+1)$$
(
k
+
1
)
-ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting formula, where we count trees with a prescribed number of labels of each kind. Several corollaries are derived from this formula, and an analogous theorem is proven for
k
-
noncrossing trees
, a similarly defined family of labelled noncrossing trees that are related to
$$(2k+1)$$
(
2
k
+
1
)
-ary trees. |
|---|---|
| ISSN: | 0218-0006 0219-3094 |
| DOI: | 10.1007/s00026-023-00642-6 |