A390353 Integrals of stable envelopes of fixed points in T^*Gr(2,n).

1, 1, 2, 1, 1, 3, 2, 2, 3, 1, 1, 4, 5, 2, 4, 10, 5, 4, 4, 1, 1, 5, 9, 7, 2, 7, 23, 22, 7, 12, 23, 9, 7, 5, 1, 1, 6, 14, 16, 9, 2, 11, 44, 61, 38, 9, 30, 80, 61, 16, 30, 44, 14, 11, 6, 1, 1, 7, 20, 30, 25, 11, 2, 16, 75, 135, 124, 58, 11, 65, 215, 240, 124, 25, 100, 215, 135, 30, 65, 75, 20, 16, 7, 1

Offset

1,3

Comments

a(n) is the coefficient of h^(2*(N-2)) in the C^{*}_{h} integral of Stab(p_{i1,i2}) over T^{*}Gr(2,N) where N is the integer such that binomial(N,3) < n <= binomial(N+1,3), and i1 and i2 are the unique integers 1 <= i1 < i2 <= N such that N*(i1 - 1) - binomial(i1,2) + (i2 - i1) = n - binomial(N,3).

This list is best thought of as arranged in a tetrahedral array; that is, as a sequence of triangular arrays. The first layer, or tip, of the tetrahedron contains 1 number, the next contains 3, then 6, etc. The triangular layers are arranged in lexicographic order by fixed point; that is, the entry corresponding to the triple {N,i1,i2} is listed before the entry {M,j1,j2} if and only if N<M or N=M and i1<j1 or N=M and i1=j1 and i2<j2 (see link).

In the diagram below, in each specific layer, the triangles are read from bottom-to-top, and then right-to-left.

This list is a generalization of Pascal's pyramid, but with an augmented 4-neighbor addition, rather than the typical 3-neighbor addition (see link).

Links

Matthew Crawford, Table of n, a(n) for n = 1..1140

Matthew Crawford, Pavan Kartik, and Reese Lance, Integrals of stable envelopes for cotangent bundles to Grassmannians, arXiv preprint arXiv:2510.21573 [math.AG], 2025.

Formula

a(n) = ((i2 + i1^2*(-1 + N) + i2^2*(-1 + N) + 2*N + i2*(-4 + N)*N + i1*(-3 - 2*i2*(-2 + N) - (-2 + N)*N))*(-3 + N)!*(-2 + N)!)/((-1 + i1)!*(-1 + i2)!*(-i1 + N)!*(-i2 + N)!) where N is the unique integer such that binomial(N,3) < n <= binomial(N+1,3), and i1 and i2 are the unique integers 1 <= i1 < i2 <= N such that N*(i1 - 1) - binomial(i1,2) + (i2 - i1) = n - binomial(N,3).

Example

For n = 8, a(8) = 2 because the coefficient of h^(2*(4-2)) in the C^{*}_{h} integral of Stab(p_{2,3}) over T^{*}Gr(2,4), and 8 = binomial(4,3) + 4*(2-1) - binomial(2,2) + (3-2).
For n = 16, a(16) = 10 because the coefficient of h^(2*(5-2)) in the C^{*}_{h} integral of Stab(p_{2,4}) over T^{*}Gr(2,5), and 16 = binomial(5,3) + 5*(2-1) - binomial(2,2) + (4-2).
Since this is a tetrahedral list, we can arrange the list as a sequence of triangular lists as shown below.
  1;
  -----------
  2;
  1, 1;
  -----------
  2;
  3, 3;
  1, 2, 1;
  -----------
  2;
  5, 5;
  4, 10, 4;
  1, 4, 4, 1;
  -----------
  ...

Mathematica

Prepend[Flatten[Table[((i2 + i1^2 (-1 + n) + i2^2 (-1 + n) + 2 n + i2 (-4 + n) n + i1 (-3 - 2 i2 (-2 + n) - (-2 + n) n)) (-3 + n)! (-2 + n)!)/((-1 + i1)! (-1 + i2)! (-i1 + n)! (-i2 + n)!), {n, 3, 8}, {i1, 1, n - 1}, {i2, i1 + 1, n}]], 1]

Crossrefs

Sequence in context: A111867 A326036 A133776 * A060118 A329143 A219032

Adjacent sequences: A390350 A390351 A390352 * A390354 A390355 A390356

Keyword

nonn,tabf

Author

Matthew Crawford, Nov 01 2025

Status

approved