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A180349 Gog words avoiding the subpattern 312.

%I A180349 #27 Sep 30 2024 12:26:22
%S A180349 1,2,6,26,162,1450,18626,343210,9069306,343611106,18662952122,
%T A180349 1453016097506,162144482866166,25932885879826066
%N A180349 Gog words avoiding the subpattern 312.
%C A180349 Gog words of size n are words of length n in an alphabet of odd-sized tuples of increasing integers that satisfy the following conditions:
%C A180349 (1) The length of the word is n,
%C A180349 (2) each letter in the word has maximum entry at most n,
%C A180349 (3) an integer in an even-numbered position in a tuple is repeated in another tuple to its left and to its right in odd-numbered positions,
%C A180349 (4) every repeated integer alternates in odd- and even-numbered positions in subsequent tuples.
%C A180349 They are in natural bijection with alternating sign matrices.
%C A180349 Further, the integers c, a, b form a 312-subpattern of the Gog word w = x_1 x_2 ... x_n if the following conditions hold:
%C A180349 (1) c, a, b appear in odd positions in x_i, x_j, x_k, respectively, where i < j < k,
%C A180349 (2) b is not in an even position in x_(i+1), ..., x_(k-1),
%C A180349 (3) if x_j = (p_1, q_1, ..., p_(k-1), q_(k-1), p_k), either b > p_k or p_l < b < q_l for some l.
%C A180349 (4) a < b < c.
%C A180349 a(n) is equal to the number of gapless Gog triangles of size n, and also to the number of gapless Magog triangles of size n. - _Ludovic Schwob_, May 18 2024
%H A180349 Arvind Ayyer, Robert Cori, and Dominique Gouyou-Beauchamps, <a href="http://arxiv.org/abs/1101.1666">Monotone triangles and 312 pattern avoidance</a>, arXiv:1101.1666 [math.CO], 2011.
%H A180349 Mathilde Bouvel, Rebecca Smith, and Jessica Striker, <a href="https://arxiv.org/abs/2408.05311">Key-avoidance for alternating sign matrices</a>, arXiv:2408.05311 [math.CO], 2024. See p. 4.
%H A180349 Ludovic Schwob, <a href="/A180349/a180349.txt">Sage program</a>.
%e A180349 For n=3, there are 7 Gog words: (1)(2)(3), (1)(3)(2), (2)(1)(3), (2)(3)(1), (3)(1)(2), (3)(2)(1) and (2)(123)(2). Of these, all but (3)(1)(2) avoid the subpattern 312.
%e A180349 More complicated examples: 31(234)3 and 25(12356)542 contain the subpattern 312 but 25(12456)532 does not.
%Y A180349 Cf. A005130, A000108, A116715, A116722, A116735.
%K A180349 nonn,hard
%O A180349 1,2
%A A180349 _Arvind Ayyer_, Jan 18 2011
%E A180349 a(13)-a(14) from _Ludovic Schwob_, May 18 2024