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Register a new userA bijection for two sequences in OEIS
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David Callan
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We show that sequences A026737 and A111279 in The On-Line Encyclopedia of Integer Sequences are the same by giving a bijection between two classes of Grand Schroder paths.
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We give a combinatorial interpretation in terms of bicolored ordered trees for the sequence (a_n)_{n>=1}=(1, 1, 1, 2, 3, 6, 10, 20, 36, 73,... ), A345973 in OEIS, whose generating function satisfies the defining identity Sum_{n>=1}a_n x^n = x + x^2/Product_{n>=1}(1 - a_n x^n).
There is a bijection from Schroder paths to {4132, 4231}-avoiding permutations due to Bandlow, Egge, and Killpatrick that sends area to inversion number. Here we give a concise description of this bijection.
For each positive integer $n$, we construct a bijection between the odd partitions and the distinct partitions of $n$ which extends Bressouds bijection between the odd-and-distinct partitions of $n$ and the splitting partitions of $n$. We compare our bijection with the classical bijections of Glaisher and Sylvester, and also with a recent bijection due to Chen, Gao, Ji and Li.
A di-sk tree is a rooted binary tree whose nodes are labeled by $oplus$ or $ominus$, and no node has the same label as its right child. The di-sk trees are in natural bijection with separable permutations. We construct a combinatorial bijection on di-sk trees proving the two quintuples $(LMAX,LMIN,DESB,iar,comp)$ and $(LMAX,LMIN,DESB,comp,iar)$ have the same distribution over separable permutations. Here for a permutation $pi$, $LMAX(pi)/LMIN(pi)$ is the set of values of the left-to-right maxima/minima of $pi$ and $DESB(pi)$ is the set of descent bottoms of $pi$, while $comp(pi)$ and $iar(pi)$ are respectively the number of components of $pi$ and the length of initial ascending run of $pi$. Interestingly, our bijection specializes to a bijection on $312$-avoiding permutations, which provides (up to the classical {em Knuth--Richards bijection}) an alternative approach to a result of Rubey (2016) that asserts the two triples $(LMAX,iar,comp)$ and $(LMAX,comp,iar)$ are equidistributed on $321$-avoiding permutations. Rubeys result is a symmetric extension of an equidistribution due to Adin--Bagno--Roichman, which implies the class of $321$-avoiding permutations with a prescribed number of components is Schur positive. Some equidistribution results for various statistics concerning tree traversal are presented in the end.
We verify the conjecture that the sixth binary partition function is equal (aside from the initial zero term) to the partial sums of the Stern-Brocot sequence.
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