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A347976 Triangle T(n,k) read by rows: the rows list volumes of rank 2 Schubert matroid polytopes.

%I A347976 #60 Mar 22 2023 17:56:57
%S A347976 1,2,4,3,8,11,4,13,22,26,5,19,38,52,57,6,26,60,94,114,120,7,34,89,158,
%T A347976 213,240,247,8,43,126,251,376,459,494,502,9,53,172,381,632,841,960,
%U A347976 1004,1013,10,64,228,557,1018,1479,1808,1972,2026,2036,11,76,295,789,1580,2503,3294,3788,4007,4072,4083
%N A347976 Triangle T(n,k) read by rows: the rows list volumes of rank 2 Schubert matroid polytopes.
%C A347976 T(n,k) is the volume of the base polytope of the Lattice Path Matroid bounded by the paths L = (n-2)*[0]+[1,1] and U = [1]+(n-k-2)*[0]+[1]+(k)*[0].
%H A347976 Carolina Benedetti, Kolja Knauer, and Jerónimo Valencia-Porras, <a href="https://arxiv.org/abs/2303.10458">On lattice path matroid polytopes: alcoved triangulations and snake decompositions</a>, arXiv:2303.10458 [math.CO], 2023.
%F A347976 T(n,k-1) + T(n,k) + k = T(n+1,k).
%F A347976 For a fixed k, the column T(n,k) is given by a polynomial in n.
%F A347976 For any 1 <= k <= n-3, T(n,k) + T(n,n-k-2) = T(n,n-2).
%e A347976 The triangle T(n,k) starts as follows:
%e A347976 [n\k] [1] [2]  [3]   [4]   [5]   [6]   [7]   [8]   [9]  [10]  [11]  [12]
%e A347976 [3]    1;
%e A347976 [4]    2,  4;
%e A347976 [5]    3,  8,  11;
%e A347976 [6]    4, 13,  22,   26;
%e A347976 [7]    5, 19,  38,   52,   57;
%e A347976 [8]    6, 26,  60,   94,  114,  120;
%e A347976 [9]    7, 34,  89,  158,  213,  240,  247;
%e A347976 [10]   8, 43, 126,  251,  376,  459,  494,  502;
%e A347976 [11]   9, 53, 172,  381,  632,  841,  960, 1004, 1013;
%e A347976 [12]  10, 64, 228,  557, 1018, 1479, 1808, 1972, 2026, 2036;
%e A347976 [13]  11, 76, 295,  789, 1580, 2503, 3294, 3788, 4007, 4072, 4083;
%e A347976 [14]  12, 89, 374, 1088, 2374, 4089, 5804, 7090, 7804, 8089, 8166, 8178;
%e A347976 ...
%Y A347976 Columns: A000027 (k=1), A034856 (k=2).
%Y A347976 Diagonals: A000295 (k=n-2), A005803 (k=n-3), A277411 (k=n-4).
%K A347976 nonn,tabl
%O A347976 3,2
%A A347976 _Jerónimo Valencia Porras_, Sep 21 2021