A174560 -id:A174560 - cp's OEIS frontend

A174561 Number of 3 X n Latin rectangles whose second row contains two cycles with the same order of its elements, e.g., the cycle (x_2, x_3, ..., x_k, x_1) with x_1 < x_2 < ... < x_k.

Original entry on oeis.org

12, 120, 2020, 32410, 563948

Offset: 4

Vladimir Shevelev, Mar 22 2010

V. S. Shevelev, Reduced Latin rectangles and square matrices with equal row and column sums, Diskr. Mat.(J. of the Akademy of Sciences of Russia) 4(1992), 91-110.

Cf. A000179, A000186, A094047, A174560, A174556.

A174563 Number of 3 X n Latin rectangles such that every element of the second row has the same cyclic order (see comment).

Original entry on oeis.org

1, 14, 133, 3300, 93889, 3391086, 148674191, 7796637196, 480640583751, 34370030511334, 2818294139246649, 262403744798653716, 27506121212584723373, 3222018028986227724702, 418998630100386520363619, 60138044879434564251209580, 9477043948863636836099726259, 1632099068624734991723488992214

Offset: 3

Vladimir Shevelev, Mar 22 2010

nonn

We say that an element alpha_i of a permutation alpha of {1,2,...,n} has cyclic order k if it belongs to a cycle of length k of alpha. If every cycle of alpha has length k, then k|n.

V. S. Shevelev, Reduced Latin rectangles and square matrices with equal row and column sums, Diskr. Mat. [Journal published by the Academy of Sciences of Russia], 4 (1992), 91-110.
V. S. Shevelev, Modern enumeration theory of permutations with restricted positions, Diskr. Mat., 1993, 5, no.1, 3-35 (Russian) [English translation in Discrete Math. and Appl., 1993, 3:3, 229-263 (pp. 255-257)].

Cf. A000179, A000186, A000296, A094047, A174556, A174560, A174561.

Let G_n = A000296(n) = n! * Sum_{2*k_2+...+n*k_n=n, k_i>=0} Product_{i=2,...,n} (k_i!*i!^k_i)^(-1). Then a(n) = Sum_{k=0,...,floor(n/2)} binomial(n,k) * G_k * G_(n-k) * u_(n-2*k), where u(n) = A000179(n). - Vladimir Shevelev, Mar 30 2016

More terms from William P. Orrick, Jul 25 2020

A176901 Number of 3 X n semireduced Latin rectangles, that is, having exactly n fixed points in the first two rows.

Original entry on oeis.org

4, 72, 1584, 70720, 3948480, 284570496, 25574768128, 2808243910656, 369925183388160, 57585548812887040, 10458478438093154304, 2191805683821733404672, 525011528578874444283904, 142540766765931981615759360, 43542026550306796238178877440, 14867182204795857282384287236096, 5640920219495105293649671985430528

Offset: 3

Vladimir Shevelev, Apr 28 2010

nonn

A Latin rectangle is called reduced if its first row is [1,2,...,n] (the number of 3 X n reduced Latin rectangles is given in A000186). Therefore a Latin rectangle having exactly n fixed points in the first two rows may be called "semireduced". Thus if A1(i), A2(i), i=1,...,n, are the first two rows, then, for every i, either A1(i)=i or A2(i)=i.

V. S. Shevelev, Reduced Latin rectangles and square matrices with equal row and column sums, Diskr. Mat.(J. of the Akademy of Sciences of Russia) 4(1992), 91-110.
V. S. Shevelev, Modern enumeration theory of permutations with restricted positions, Diskr. Mat., 1993, 5, no.1, 3-35 (Russian).
V. S. Shevelev, Modern enumeration theory of permutations with restricted positions, English translation, Discrete Math. and Appl., 1993, 3:3, 229-263 (pp. 255-257).

Cf. A174563, A000179, A000186, A087981, A094047, A174556, A174560, A174561.

Let F_n = A087981(n) = n! * Sum_{2*k_2+...+n*k_n=n, k_i>=0} Product_{i=2..n} 2^k_i/(k_i!*i^k_i). Then a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * F_k * F_(n-k) * u_(n-2*k), where u(n) = A000179(n). - Vladimir Shevelev, Mar 30 2016

More terms from William P. Orrick, Jul 25 2020

cp's OEIS Frontend

A174561 Number of 3 X n Latin rectangles whose second row contains two cycles with the same order of its elements, e.g., the cycle (x_2, x_3, ..., x_k, x_1) with x_1 < x_2 < ... < x_k.

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A174563 Number of 3 X n Latin rectangles such that every element of the second row has the same cyclic order (see comment).

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A176901 Number of 3 X n semireduced Latin rectangles, that is, having exactly n fixed points in the first two rows.

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