%I #30 Sep 26 2023 22:56:06
%S 1,1,4,18,124,738,7464,55890,668778,7030210,90713844,1054221258,
%T 18597735744,242795838520
%N The number of permutations of {1,1,1,2,2,2,...,n,n,n} such that each triple of k's (k=1..n) is equally spaced with b(k) other elements in between and b(1) >= b(2) >= ... >= b(n).
%e n = 1 case:
%e | | b(1)
%e -----+-----------+------
%e 1 | [1, 1, 1] | [0] *
%e .
%e n = 2 case:
%e | | b(1),b(2)
%e -----+--------------------+----------
%e 1 | [2, 2, 2, 1, 1, 1] | [0, 0]
%e 2 | [2, 1, 2, 1, 2, 1] | [1, 1]
%e 3 | [1, 2, 1, 2, 1, 2] | [1, 1]
%e 4 | [1, 1, 1, 2, 2, 2] | [0, 0]
%e .
%e n = 3 case:
%e | | b(1),b(2),b(3)
%e -----+-----------------------------+---------------
%e 1 | [3, 3, 3, 2, 2, 2, 1, 1, 1] | [0, 0, 0]
%e 2 | [3, 3, 3, 2, 1, 2, 1, 2, 1] | [1, 1, 0]
%e 3 | [3, 3, 3, 1, 2, 1, 2, 1, 2] | [1, 1, 0]
%e 4 | [3, 3, 3, 1, 1, 1, 2, 2, 2] | [0, 0, 0]
%e 5 | [3, 2, 1, 3, 2, 1, 3, 2, 1] | [2, 2, 2]
%e 6 | [3, 1, 2, 3, 1, 2, 3, 1, 2] | [2, 2, 2]
%e 7 | [1, 3, 3, 3, 1, 2, 2, 2, 1] | [3, 0, 0]
%e 8 | [2, 3, 1, 2, 3, 1, 2, 3, 1] | [2, 2, 2]
%e 9 | [1, 3, 2, 1, 3, 2, 1, 3, 2] | [2, 2, 2]
%e 10 | [2, 1, 3, 2, 1, 3, 2, 1, 3] | [2, 2, 2]
%e 11 | [1, 2, 3, 1, 2, 3, 1, 2, 3] | [2, 2, 2]
%e 12 | [2, 2, 2, 3, 3, 3, 1, 1, 1] | [0, 0, 0]
%e 13 | [1, 1, 1, 3, 3, 3, 2, 2, 2] | [0, 0, 0]
%e 14 | [1, 2, 2, 2, 1, 3, 3, 3, 1] | [3, 0, 0]
%e 15 | [2, 2, 2, 1, 1, 1, 3, 3, 3] | [0, 0, 0]
%e 16 | [2, 1, 2, 1, 2, 1, 3, 3, 3] | [1, 1, 0]
%e 17 | [1, 2, 1, 2, 1, 2, 3, 3, 3] | [1, 1, 0]
%e 18 | [1, 1, 1, 2, 2, 2, 3, 3, 3] | [0, 0, 0]
%e * (strongly decreasing)
%Y Column k=3 of A332762.
%Y Cf. A104429, A059108, A261516 (strongly decreasing), A322178, A332752.
%K nonn,more
%O 0,3
%A _Seiichi Manyama_, Feb 21 2020
%E a(10)-a(13) from _Max Alekseyev_, Sep 26 2023