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A347976
Triangle T(n,k) read by rows: the rows list volumes of rank 2 Schubert matroid polytopes.
0
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
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: A377835 A368221 A077632 * A387024 A253722 A361640
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved