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Section 1 introduces numerous definitions and notations, and most notably the tree-like tableaux which are the central focus of this work.
Alternative tableaux and permutation tableaux: Tree-like tableaux are closely related to alternative tableaux [Nad09, Vie08] as follows: given a tree-like tableau, change every non-root point p to an arrow which is oriented left (respectively up) if there is no point left of p (resp.
1) Tree-like tableaux of half-perimeter n + 1, with i left points, j top points, k rows, l crossings.
2) Permutation tableaux of half-perimeter n with i + 1 unrestricted rows, j top ones, k - 1 rows, l superfluous ones.
3) Alternative tableaux of half-perimeter n - 1 with i free rows, j free columns, k - 1 rows, l free cells.
We describe a new way of inserting points in tree-like tableaux, shedding new light on numerous enumerative results on those tableaux.
So we have an elementary proof that tableaux of size n are equinumerous with permutations of length n.
tree-like tableaux which are invariant with respect to reflection through the main diagonal of their diagram.
It is easy to check that when only [epsilon] = +1 is chosen during insertions, then the symmetric tableaux obtained in this case are precisely those corresponding to the embedding of usual tree-like tableaux into symmetric ones defined above.
We now show how our insertion procedures give elementary proofs of some enumerative results on tableaux.
We can also give a generalization of Formula (1) to symmetric tableaux [LW08, CK10].
This was proved first in [CH07, Theorem 1] by a lengthy computation, which relied on the recursive construction of (permutation) tableaux obtained by adding the leftmost column.
We will here define one with the goal of sending crossings of tableaux to occurrences of the pattern 2-31 in a permutation.