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

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, 213, 240, 247, 8, 43, 126, 251, 376, 459, 494, 502, 9, 53, 172, 381, 632, 841, 960, 1004, 1013, 10, 64, 228, 557, 1018, 1479, 1808, 1972, 2026, 2036, 11, 76, 295, 789, 1580, 2503, 3294, 3788, 4007, 4072, 4083

Offset: 3

Author

Jerónimo Valencia Porras, Sep 21 2021

Keywords

• nonn
• tabl

Comments

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].

Examples

			The triangle T(n,k) starts as follows:
[n\k] [1] [2]  [3]   [4]   [5]   [6]   [7]   [8]   [9]  [10]  [11]  [12]
[3]    1;
[4]    2,  4;
[5]    3,  8,  11;
[6]    4, 13,  22,   26;
[7]    5, 19,  38,   52,   57;
[8]    6, 26,  60,   94,  114,  120;
[9]    7, 34,  89,  158,  213,  240,  247;
[10]   8, 43, 126,  251,  376,  459,  494,  502;
[11]   9, 53, 172,  381,  632,  841,  960, 1004, 1013;
[12]  10, 64, 228,  557, 1018, 1479, 1808, 1972, 2026, 2036;
[13]  11, 76, 295,  789, 1580, 2503, 3294, 3788, 4007, 4072, 4083;
[14]  12, 89, 374, 1088, 2374, 4089, 5804, 7090, 7804, 8089, 8166, 8178;
...
		

Links

• Carolina Benedetti, Kolja Knauer, and Jerónimo Valencia-Porras, On lattice path matroid polytopes: alcoved triangulations and snake decompositions, arXiv:2303.10458 [math.CO], 2023.

Crossrefs

Columns: A000027 (k=1), A034856 (k=2).

Diagonals: A000295 (k=n-2), A005803 (k=n-3), A277411 (k=n-4).

Formula

T(n,k-1) + T(n,k) + k = T(n+1,k).

For a fixed k, the column T(n,k) is given by a polynomial in n.

For any 1 <= k <= n-3, T(n,k) + T(n,n-k-2) = T(n,n-2).