A349458 -id:A349458 - cp's OEIS frontend

A349413 Number of smooth positroid varieties corresponding to derangements in S_n.

1, 0, 1, 2, 5, 14, 40, 118, 357, 1100

Offset: 0

Jordan Weaver, Nov 16 2021

a(n) is also the number of derangements in S_n whose chordal diagram contains no crossed alignments.
a(n) is also the number of derangements in S_n whose chordal diagram is a separable union of star graphs, where a star graph is the chordal diagram of a permutation in S_m of the form w(i) = i + t (mod m) for some t.
a(n) counts the complement of A349456 in the set of all derangements of S_n (A000166).
a(n) appears to be the number of n-edge ordered trees in which each nonleaf has at least two children and each leftmost child has a designated favorite sibling. For example, for n = 3, the underlying tree must be a root with 3 children and there are two choices for the favorite sibling, so a(3) = 2. The generating function for these trees, A(x) = 1 + x^2 + 2*x^3 + 5*x^4 + ..., is easily shown, using the "symbolic method" of Flajolet and Sedgewick, to satisfy A(x) = 1 + x^2*A(x)^2/(1 - x*A(x))^2. - David Callan, May 15 2022

			For n=4, the a(4)=5 derangements in one-line notation are 2143, 4321, 2341, 4123, and 3412.

Sara C. Billey and Jordan E. Weaver, Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs, arXiv:2207.06508 [math.CO], 2022.
S. Corteel, Crossings and alignments of permutations, arXiv:math/0601469 [math.CO], 2006.
A. Knutson, T. Lam and D. Speyer, Positroid varieties: juggling and geometry, Compos. Math. 149 (2013), no. 10, 1710-1752.
A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764 [math.CO], 2006.

Cf. A000166, A349456, A349458, A349457.

a(n) = A000166(n) - A349456(n).

A349456 Number of singular positroid varieties corresponding to derangements in S_n.

Original entry on oeis.org

0, 0, 0, 0, 4, 30, 225, 1736, 14476, 132396

Offset: 0

Jordan Weaver, Nov 16 2021

a(n) is also the number of derangements whose chordal diagrams have crossed alignments.

a(n) counts the complement of A349413 in the set of all derangements of S_n (A000166).

			For n=4 the a(4)=4 derangements in one-line notation corresponding to singular positroid varieties are 2413, 3421, 3142, and 4312.

Sara C. Billey and Jordan E. Weaver, Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs, arXiv:2207.06508 [math.CO], 2022.
S. Corteel, Crossings and alignments of permutations, arXiv:math/0601469 [math.CO], 2006.
A. Knutson, T. Lam and D. Speyer, Positroid varieties: juggling and geometry, Compos. Math. 149 (2013), no. 10, 1710-1752.
A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764 [math.CO], 2006.

Cf. A000166, A349413, A349457, A349458.

a(n) = A000166(n) - A349413(n).

A349457 Number of singular positroids in the Grassmannian variety Gr(k,n) for a fixed n and any 0 <= k <= n.

Original entry on oeis.org

0, 0, 0, 0, 4, 70, 825, 8526, 85372, 870756

Offset: 0

Jordan Weaver, Nov 17 2021

a(n) is also the number of decorated permutations whose chordal diagram contains a crossed alignment.

a(n) counts the complement of A349458 in the set of all positroid varieties/decorated permutations on n elements (A000522).

			For n = 4, the a(4) = 4 singular positroid varieties correspond to the decorated permutations whose underlying permutations are 2413, 3421, 3142, and 4312 in one-line notation. Note that none of these permutations contain fixed points, hence no decorations are needed.

Sara C. Billey and Jordan E. Weaver, Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs, arXiv:2207.06508 [math.CO], 2022.
S. Corteel, Crossings and alignments of permutations, arXiv:math/0601469 [math.CO], 2006.
A. Knutson, T. Lam and D. Speyer, Positroid varieties: juggling and geometry, Compos. Math. 149 (2013), no. 10, 1710-1752.
A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764 [math.CO], 2006.

Cf. A000522, A349413, A349456, A349458.

a(n) = Sum_{i=0..n} (2^i)*binomial(n,i)*b(n), where b(n) is the sequence A349456.

a(n) = A000522(n) - A349458(n).

A353131 Triangle read by rows of partial Bell polynomials B_{n,k} (x_1,...,x_{n-k+1}) evaluated at 2, 2, 12, 72, ..., (n-k)(n-k+1)!, n>=1, 1<=k<=n.

Original entry on oeis.org

2, 2, 4, 12, 12, 8, 72, 108, 48, 16, 480, 960, 600, 160, 32, 3600, 9360, 7320, 2640, 480, 64, 30240, 100800, 95760, 42000, 10080, 1344, 128, 282240, 1189440, 1350720, 700560, 201600, 34944, 3584, 256, 2903040, 15240960, 20442240, 12337920, 4142880, 854784, 112896, 9216, 512

Offset: 1

Jordan Weaver, Apr 24 2022

			For n=4,k=2, the partial Bell polynomial is B_{4,2}(x_1,x_2,x_3)=4x_1x_3+3x_2^2, so a(4,2)=B_{4,2}(2,2,12)=4*2*12+3*2^2=108.
Triangle starts:
[1]      2;
[2]      2,       4;
[3]     12,      12,       8;
[4]     72,     108,      48,     16;
[5]    480,     960,     600,    160,     32;
[6]   3600,    9360,    7320,   2640,    480,    64;
[7]  30240,  100800,   95760,  42000,  10080,  1344,  128;
[8] 282240, 1189440, 1350720, 700560, 201600, 34944, 3584, 256.

Jordan Weaver, Rows 1 to 40 of triangle, flattened
E. T. Bell, Partition polynomials, Ann. Math., 29 (1927-1928), 38-46.
E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934), 258-277.
Sara C. Billey and Jordan E. Weaver, Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs, arXiv:2207.06508 [math.CO], 2022.
A. Knutson, T. Lam and D. Speyer, Positroid varieties: juggling and geometry, Compos. Math. 149 (2013), no. 10, 1710-1752.
A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764 [math.CO], 2006.

Cf. A000079, A353132, A349413, A268441, A178867.

T(n,k) = A353132(n,k)*(n-k+1)!.

Sum_{k=1..n} T(n,k)/(n-k+1)! = A349458(n).

A353132 Triangle read by rows of partial Bell polynomials B_{n,k}(x_1,...,x_{n-k+1}) evaluated at 2, 2, 12, 72, ..., (n-k)(n-k+1)!, divided by (n-k+1)!, n >= 1, 1 <= k <= n.

Original entry on oeis.org

2, 1, 4, 2, 6, 8, 3, 18, 24, 16, 4, 40, 100, 80, 32, 5, 78, 305, 440, 240, 64, 6, 140, 798, 1750, 1680, 672, 128, 7, 236, 1876, 5838, 8400, 5824, 1792, 256, 8, 378, 4056, 17136, 34524, 35616, 18816, 4608, 512, 9, 580, 8190, 45480, 122682, 175896, 137760, 57600, 11520, 1024

Offset: 1

Jordan Weaver, Apr 24 2022

			For n = 4, k = 2, the partial Bell polynomial is B_{4,2}(x_1,x_2,x_3) = 4*x_1*x_3 + 3*x_2^2, so T(4,2) = B_{4,2}(2,2,12) - (4*2*12 + 3*2^2)/3! = 18.
Triangle begins:
   [1] 2;
   [2] 1,   4;
   [3] 2,   6,    8;
   [4] 3,  18,   24,    16;
   [5] 4,  40,  100,    80,     32;
   [6] 5,  78,  305,   440,    240,     64;
   [7] 6, 140,  798,  1750,   1680,    672,    128;
   [8] 7, 236, 1876,  5838,   8400,   5824,   1792,   256;
   [9] 8, 378, 4056, 17136,  34524,  35616,  18816,  4608,   512;
  [10] 9, 580, 8190, 45480, 122682, 175896, 137760, 57600, 11520, 1024.

Jordan Weaver, Rows 1 to 40 of triangle, flattened
E. T. Bell, Partition polynomials, Ann. Math., 29 (1927-1928), 38-46.
E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934), 258-277.
Sara C. Billey and Jordan E. Weaver, Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs, arXiv:2207.06508 [math.CO], 2022.
A. Knutson, T. Lam and D. Speyer, Positroid varieties: juggling and geometry, Compos. Math. 149 (2013), no. 10, 1710-1752.
A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764 [math.CO], 2006.

Cf. A000079, A353131, A349413, A268441, A178867.

T(n,k) = A353131(n,k)/(n-k+1)!

Sum_{k=1..n} T(n,k) = A349458(n).

cp's OEIS Frontend

A349413 Number of smooth positroid varieties corresponding to derangements in S_n.

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A349456 Number of singular positroid varieties corresponding to derangements in S_n.

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A349457 Number of singular positroids in the Grassmannian variety Gr(k,n) for a fixed n and any 0 <= k <= n.

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A353131 Triangle read by rows of partial Bell polynomials B_{n,k} (x_1,...,x_{n-k+1}) evaluated at 2, 2, 12, 72, ..., (n-k)(n-k+1)!, n>=1, 1<=k<=n.

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A353132 Triangle read by rows of partial Bell polynomials B_{n,k}(x_1,...,x_{n-k+1}) evaluated at 2, 2, 12, 72, ..., (n-k)(n-k+1)!, divided by (n-k+1)!, n >= 1, 1 <= k <= n.

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