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A330490 Total number of permutation arrays of side length n and dimension k as defined by Eriksson and Linusson (2000a); square array T(n,k), read by ascending antidiagonals, for n, k >= 1. 1

%I #24 Dec 17 2019 02:39:50

%S 1,1,1,1,2,1,1,6,5,1,1,24,70,15,1,1,120,2167,1574,52,1,1,720,130708,

%T 968162,69874,203,1,1,5040,14231289

%N Total number of permutation arrays of side length n and dimension k as defined by Eriksson and Linusson (2000a); square array T(n,k), read by ascending antidiagonals, for n, k >= 1.

%C The poset P_{3 x 3} of (3 x 3 x 3)-permutation arrays is shown in Figure 1 on p. 209 of Eriksson and Linuson (2000a). We have |P_{3 x 3}| = T(3,3) = 70. The numbers in this rectangular array are copied from Table 1 (p. 210) of the same paper.

%H Kimmo Eriksson and Svante Linusson, <a href="https://doi.org/10.1006/aama.1999.0693">A combinatorial theory of higher-dimensional permutation arrays</a>, Adv. Appl. Math. 25(2) (2000a), 194-211.

%H Kimmo Eriksson and Svante Linusson, <a href="https://doi.org/10.1006/aama.2000.0694">A decomposition of Fl(n)^d indexed by permutation arrays</a>, Adv. Appl. Math. 25(2) (2000b), 212-227. [Fl(n)^d denotes the flag manifold over C^n.]

%F T(n=1,k) = 1 = A000012(n) and T(n=2,k) = A000110(k) (Bell numbers).

%F T(n,k=1) = 1 = A000012(n) and T(n,k=2) = n! = A000142(n).

%F T(n,k) >= (n!)^(k-1) = A225816(k-1, n).

%F T(n,k=3) <= n!*2^(binomial(n+1,2) - 1).

%e Array T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows, where * indicates a missing number:

%e 1, 1, 1, 1, 1, ...

%e 1, 2, 5, 15, 52, ...

%e 1, 6, 70, 1574, 69874, ...

%e 1, 24, 2167, 968162, *, ...

%e 1, 120, 130708, *, *, ...

%e 1, 720, 14231289, *, *, ...

%e 1, 5040, 2664334184, *, *, ...

%e 1, 40320, 831478035698, *, *, ...

%e ...

%Y Cf. A000012, A000110, A000142, A225816.

%K nonn,tabl,more

%O 1,5

%A _Petros Hadjicostas_, Dec 16 2019

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