The Raney numbers $R_{p,r}(n)$ are a two-parameter generalization of the Catalan numbers that were introduced by Raney in his investigation of functional composition patterns \cite{Raney}. We give a new combinatorial interpretation for... more
The Raney numbers $R_{p,r}(n)$ are a two-parameter generalization of the Catalan numbers that were introduced by Raney in his investigation of functional composition patterns \cite{Raney}. We give a new combinatorial interpretation for all Raney numbers in terms of planar embeddings of certain collections of trees, a construction that recovers the usual interpretation of the $p$-Catalan numbers in terms of $p$-ary trees via the specialization $R_{p,1}(n) =_{p} c_n$. Our technique leads to several combinatorial identities involving the Raney numbers and ordered partitions. We then give additional combinatorial interpretations of specific Raney numbers, including an identification of $R_{p^2,p}(n)$ with oriented trees whose vertices satisfy the "source or sink property". We close with comments applying these results to the enumeration of connected (non-elliptic) $A_2$ webs that lack an internal cycle.
A tableaux inversion is a pair of entries in row-standard tableaux $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard. An $i$-inverted Young tableaux is a row-standard tableaux... more
A tableaux inversion is a pair of entries in row-standard tableaux $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard. An $i$-inverted Young tableaux is a row-standard tableaux along with a precisely $i$-inversion pairs. Tableaux inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of $i$-inverted tableaux that standardize to a fixed standard Young tableaux corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableaux inversions from a completely combinatorial perspective. We develop formulas enumerating the number of $i$-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableaux that standardize a specific standard Young tableaux, and construct bijections between $i$-inverted Young tableaux of a certain shape with $j$-inverted Young tableaux of different sh...
A tableau inversion is a pair of entries in row-standard tableau $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard. An $i$-inverted Young tableau is a row-standard tableau along... more
A tableau inversion is a pair of entries in row-standard tableau $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard. An $i$-inverted Young tableau is a row-standard tableau along with precisely $i$ inversion pairs. Tableau inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of $i$-inverted tableaux that standardize to a fixed standard Young tableau corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableau inversions from a completely combinatorial perspective. We develop formulas enumerating the number of $i$-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableau that standardize a specific standard Young tableau, and construct bijections between $i$-inverted Young tableaux of a certain shape with $j$-inverted Young tableaux of different shapes. Fina...
In 1935, Paul Erdős and George Szekeres were able to show that any point set large enough contains the vertices of a convex k-gon. Later in 1961, they constructed a point set of size 2k−2 not containing the vertex set of any convex k-gon.... more
In 1935, Paul Erdős and George Szekeres were able to show that any point set large enough contains the vertices of a convex k-gon. Later in 1961, they constructed a point set of size 2k−2 not containing the vertex set of any convex k-gon. This leads to what is known as the Erdős-Szekeres Conjecture, that any point set of 2k−2 + 1 points contains the vertices of a convex k-gon. Recently, this famous problem of planar geometry has been transformed into a problem of finding cliques in a graph of copoints. We will discuss results and open problems corresponding to this graph of copoints.
A tableau inversion is a pair of entries in row-standard tableau T that lie in the same column of T yet lack the appropriate relative ordering to make T column-standard. An i-inverted Young tableau is a row-standard tableau along with a... more
A tableau inversion is a pair of entries in row-standard tableau T that lie in the same column of T yet lack the appropriate relative ordering to make T column-standard. An i-inverted Young tableau is a row-standard tableau along with a precisely i inversion pairs. Tableau inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of i-inverted tableaux that standardize to a fixed standard Young tableau corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableau inversions from a completely combinatorial perspective. We develop formulas enumerating the number of i-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableaux that standardize a specific standard Young tableau, and construct bijections between i-inverted Young tableaux of a certain shape with j-inverted Young tableaux of different shapes. Finally, we share so...
