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A rack is involutory if it satisfies the identity y(yx) = x. In particular, involutory quandles are called kei.

a(n) is also the number of Legendrian kei (i.e., kei equipped with Legendrian structures) up to order n up to isomorphism; see Ta, Theorem 1.1.

a(n) is also the number of symmetric kei (i.e., kei equipped with good involutions) up to order n up to isomorphism; see Ta, "Equivalences of...," Corollary 1.3.

An m X n link mosaic is a suitably connected m X n array of the 11 tiles given by Lomonaco and Kauffman. The condition of being suitably connected means that the connection points of each tile coincide with those of the contiguous tiles. Thus, link mosaics depict projections of a link or a knot onto a plane.

The Mathematica program below is based on the algorithm given in Theorem 1 of Oh, Hong, Lee, and Lee.

An m X n Legendrian mosaic is an m X n array of the 10 tiles given in Figure 5 of Pezzimenti and Pandey. These tiles represent part of a Legendrian curve in the front projection. The condition of being suitably connected means that the connection points of each tile coincide with those of the contiguous tiles.

The Mathematica program below is based on the algorithm given in Theorem 1 of Oh, Hong, Lee, and Lee, adapted to the Legendrian setting: since Legendrian mosaic tiles omit the crossing tile T_9 used in general knot mosaics, the bottom-right submatrix of O_(k+1) is 3*O_k rather than 4*O_k. See Theorem 6 of Kipe et al.

T(m,2) = A375353(m,2) = A000079(m-1) for all m >= 2 since neither classical nor Legendrian link mosaics with only 2 columns or rows can use T_9 tiles.

An m X n mosaic is an m X n array of the 11 tiles given by Lomonaco and Kauffman. A period m X n mosaic is an m X n mosaic whose opposite edges are identified. A period mosaic depicts a knot or link iff the connection points of each tile coincide with those of the contiguous tiles and with those of the tiles on identified edges.

The Mathematica program below is based on the algorithm given in Theorem 2 of Oh, Hong, Lee, Lee, and Yeon.

T(m, n) >= A375356(m, n) for all m and n, with equality iff m = n = 1.

T(m, 1) = A074600(m) for all m. To see this, proceed by induction on m. In Theorem 2 of Oh, Hong, Lee, Lee, and Yeon, it is clear that tr(X_{m+1}) = 2*tr(X_m) and tr(O_{m+1}) = 5*tr(O_m) for all m. The theorem states that T(m+1, 1) = tr(X_{m+1} + O_{m+1}) = tr(X_{m+1}) + tr(O_{m+1}) = 2*tr(X_m) + 5*tr(O_m), and the claim follows since tr(X_1 + O_1) = 7.

An m X n mosaic is an m X n array of the 11 tiles given by Lomonaco and Kauffman. A period m X n mosaic is an m X n mosaic whose opposite edges are identified. A toroidal m X n mosaic is an equivalence class of period m X n mosaics up to finite sequences of cyclic rotations of rows and columns. A toroidal mosaic depicts the projection of a knot or link on the surface of a torus iff the connection points of each tile coincide with those of the contiguous tiles and with those of the tiles on identified edges.

The first five rows of the triangle are from Table 2 of Oh, Hong, Lee, Lee, and Yeon.

Clearly, T(m,n) <= A375355(m,n) for all m,n, with equality iff m = n = 1.

A p X p mosaic is an p X p array of the 11 tiles given by Lomonaco and Kauffman. A period p X p mosaic is an p X p mosaic whose opposite edges are identified. A toroidal p X p mosaic is an equivalence class of period p X p mosaics up to finite sequences of cyclic rotations of rows and columns. A toroidal mosaic depicts the projection of a knot or link on the surface of a torus iff the connection points of each tile coincide with those of the contiguous tiles and with those of the tiles on identified edges.

The Mathematica program below is based on the algorithm given in Theorem 4 of Oh, Hong, Lee, Lee, and Yeon.

A rack is involutory if it satisfies the identity y(yx) = x. In particular, involutory quandles are called kei.

A rack is medial if it satisfies the identity (xy)(uv) = (xu)(yv).

a(n) is also the number of medial Legendrian kei (i.e., medial kei equipped with Legendrian structures) up to order n up to isomorphism; see Ta, Theorem 1.1.

a(n) is also the number of medial symmetric kei (i.e., medial kei equipped with good involutions) up to order n up to isomorphism; see Ta, "Equivalences of...," Corollary 1.3.

A Legendrian quandle is a pair (X,u) where X is a quandle and u is an involutory automorphism of X such that u(yx)=y(u(x)); see Ta, "Generalized Legendrian...," Corollary 3.13.

a(n) is also the number of racks X such that the kink map X -> X defined by x -> x(x) is an involution; see Ta, "Equivalences of...," Theorem 1.1.