A349413 -id:A349413 - cp's OEIS frontend

A349458 Number of smooth positroids in the Grassmannian variety Gr(k,n) for a fixed n and any 0 <= k <= n.

1, 2, 5, 16, 61, 256, 1132, 5174, 24229, 115654, 560741, 2754082, 13674212, 68522208, 346100952, 1760213254, 9006390373, 46329244034, 239455376071, 1242923653316, 6476376834789, 33863408028888, 177625109853808, 934404580376016

Offset: 0

Jordan Weaver, Nov 17 2021

nonn

a(n) is also the number of decorated permutations whose chordal diagram is a separable union of star graphs.
a(n) is also the number of decorated permutations whose chordal diagram contains no crossed alignments.
a(n) counts the complement of A349457 in the set of all positroid varieties/decorated permutations on n elements (A000522).

			For n = 3, the a(3) = 16 positroids correspond the decorated permutations with underlying permutations 231, 312, 321, 213, 132, and 123 in one-line notation. Each fixed point, e.g., the 2 in 321, can be colored in two ways. Hence 321, 213, and 132 contribute 2 decorated permutations each, 123 contributes 8, while 231 and 312 each contribute 1.

Jordan Weaver, Table of n, a(n) for n = 0..50
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, A349457.

a(n) = Sum_{i=0..n} (2^i)*binomial(n,i)*b(n), where b(n) is the sequence A349413.

a(n) = A000522(n) - A349457(n).

a(10)-a(23) from Jordan Weaver, Apr 19 2022

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

A349458 Number of smooth positroids in the Grassmannian variety Gr(k,n) for a fixed n and any 0 <= k <= 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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Formula

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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Formula