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

3,2

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].

Links

Table of n, a(n) for n=3..68.

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

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).

Example

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;
...

Crossrefs

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

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

Sequence in context: A186003 A368221 A077632 * A253722 A361640 A323506

Adjacent sequences: A347973 A347974 A347975 * A347977 A347978 A347979

Keyword

nonn,tabl

Author

Jerónimo Valencia Porras, Sep 21 2021

Status

approved