Equality of two quadrant marked mesh pattern polynomials

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Let Q132(a,b,τ,c)(t,x)Q_{132}^{(a,b,\tau,c)}(t,x) denote the polynomial associated with 132-avoiding permutations and the indicated quadrant marked mesh pattern parameters. The parameters are (0,k,∅,0)(0,k,\emptyset,0) and (1,k−1,∅,0)(1,k-1,\emptyset,0), with k≥1k\geq 1.

Equality conjecture. For all k≥1k\geq 1, we have

Q132(0,k,∅,0)(t,x)=Q132(1,k−1,∅,0)(t,x).Q_{132}^{(0,k,\emptyset,0)}(t,x)=Q_{132}^{(1,k-1,\emptyset,0)}(t,x).

The equality is motivated by the observed coincidence of the corresponding polynomials. Its resolution is not specified in the supplied text.

References

Primary source

Dun Qiu and Jeffrey B. Remmel, “Quadrant marked mesh patterns in 123-avoiding permutations”, arXiv:1705.00164 (2018).

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