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A260332 Labelings of n diamond-shaped posets with 4 vertices per diamond where the labels follow the poset relations whose associated reading permutation avoids 231 in the classical sense. 11
1, 2, 18, 226, 3298, 52450, 881970 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

According to Yang-Jiang (2021) these are the 5-Schroeder numbers. If confirmed, this will prove Michael Weiner's conjectures and enable us to extend the sequence. Yang & Jiang (2021) give an explicit formula for the m-Schroeder numbers in Theorem 2.4. - N. J. A. Sloane, Mar 28 2021

By diamond-shaped poset with 4 vertices, we mean a poset on four elements with e_1 < e_2, e_1 < e_3, e_2 < e_4, e_3 < e_4, and no order relations between e_2 and e_3. In the Hasse diagram for such a poset, we have a least element, two elements in the level above, and one element in the top level, so the diagram resembles a diamond.  The associated permutation is the permutation obtained by reading the labels of each poset by levels left to right, starting with the least element.

Also the number of labelings of n diamond-shaped posets with 4 vertices per diamond where the labels follow the poset relations whose associated reading permutation avoids 312 in the classical sense via reverse complement Wilf equivalence.

Conjecture: Also the number of lattice paths (Schroeder paths) from (0,0) to (n,4n) with unit steps N=(0,1), E=(1,0) and D=(1,1) staying weakly above the line y = 4x. - Michael D. Weiner, Jul 24 2019

REFERENCES

Sheng-Liang Yang and Mei-yang Jiang, The m-Schröder paths and m-Schröder numbers, Disc. Math. (2021) Vol. 344, Issue 2, 112209. doi:10.1016/j.disc.2020.112209. See Table 1.

LINKS

Table of n, a(n) for n=0..6.

M. Paukner, L. Pepin, M. Riehl, and J. Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015.

Manda Riehl, A 231-avoiding diamond whose associated permutation is 1234.

FORMULA

There is a complicated recursive formula available in Paukner et al.

Yang & Jiang (2021) give an explicit formula for the 5-Schroeder numbers in Theorem 2.4. - N. J. A. Sloane, Mar 28 2021

Conjecture: a(n) = Sum_{k=1..2*n} binomial(n,k)*binomial(4*n,k-1)*2^k/n for n > 0. - Michael D. Weiner, Jul 23 2019

EXAMPLE

For a single diamond (n=1) poset with 4 vertices, we must label the least element 1 and the greatest element 4, and the two central elements can be labeled either 2, 3 or 3, 2 respectively. Thus the associated permutations are 1234 and 1324.

CROSSREFS

Cf. A260331, A260579.

The sequences listed in Yang-Jiang's Table 1 appear to be A006318, A001003, A027307, A034015, A144097, A243675, A260332, A243676. - N. J. A. Sloane, Mar 28 2021

Sequence in context: A155666 A227934 A245112 * A254999 A024486 A052635

Adjacent sequences:  A260329 A260330 A260331 * A260333 A260334 A260335

KEYWORD

nonn,more

AUTHOR

Manda Riehl, Jul 29 2015

STATUS

approved

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Last modified October 18 01:39 EDT 2021. Contains 348065 sequences. (Running on oeis4.)