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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.

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).

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).

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

Original entry on oeis.org

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

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).

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).

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

Original entry on oeis.org

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).

A334156 Triangle read by rows: T(n,m) is the number of length n decorated permutations avoiding the word 0^m = 0...0 of m 0's, where 1 <= m <= n.

Original entry on oeis.org

1, 2, 4, 6, 12, 15, 24, 48, 60, 64, 120, 240, 300, 320, 325, 720, 1440, 1800, 1920, 1950, 1956, 5040, 10080, 12600, 13440, 13650, 13692, 13699, 40320, 80640, 100800, 107520, 109200, 109536, 109592, 109600, 362880, 725760, 907200, 967680, 982800, 985824, 986328, 986400, 986409

Offset: 1

Jordan Weaver, Apr 16 2020

A length n decorated permutation is a word w = w_1....w_n on the letters {0,...,n} such that the restriction of w to its nonzero entries is an ordinary permutation in one-line notation. Then w avoids 0^m if w contains at most m-1 0's as letters, and w contains 0^m if w contains m 0's among its letters (not necessarily consecutive).

			For (n,m) = (3,2), the T(3,2) = 12 length 3 decorated permutations avoiding 0^2 = 00 are 012, 102, 120, 021, 201, 210, 123, 132, 213, 231, 312, and 321.
Triangle begins:
    1
    2,   4
    6,  12,  15
   24,  48,  60,  64
  120, 240, 300, 320, 325

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
S. Corteel, Crossings and alignments of permutations, Adv. Appl. Math 38 (2007) 149-163.
A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764 [math.CO], 2006.

Cf. A000142 (1st column), A007526 (right diagonal).

Row sums are A093964.

Array[Accumulate[#!/Range[0,#-1]!]&,10] (* Paolo Xausa, Jan 08 2024 *)

T(n,m)={sum(j=0, m-1, n!/j!)} \\ Andrew Howroyd, May 11 2020

T(n,m) = Sum_{j=0..m-1} n!/j!.

Terms a(37) and beyond from Andrew Howroyd, Jan 07 2024

A334155 a(n) is the number of length n decorated permutations avoiding the pattern 001.

Original entry on oeis.org

1, 2, 5, 15, 57, 273, 1593, 10953, 86553, 771993, 7666713, 83871513, 1001957913, 12976997913, 181106559513, 2709277004313, 43247182412313, 733699248716313, 13182759232076313, 250070586344012313, 4994229502288460313, 104743211837530700313, 2301653725221036620313

Offset: 0

Jordan Weaver, Apr 16 2020

nonn

A decorated permutation of length n is a word w=w_1...w_n on the letters {0,...,n} such that the restriction of w to its nonzero entries is an ordinary permutation in one-line notation. Then w avoids the pattern 001 if there is no subword w_{i_1}w_{i_2}w_{i_3} with i_1 < i_2 < i_3 such that w_{i_1}=w_{i_2} = 0 and w_{i_3} > 0.
a(n) is also the number of decorated permutations of length n avoiding the pattern 010. This can be proved via a simple bijection mapping a 001-avoiding decorated permutation to a 010-avoiding decorated permutation.
The number of decorated permutations of length n avoiding the pattern 012 is A334154.

			For n=3, the a(3)=15 decorated permutations avoiding 001 are 000, 010, 100, 012, 102, 120, 021, 201, 210, 123, 132, 213, 231, 312, and 321.
For n=5, 10302 does not avoid 001, because it contains the subword 002.

Cf. A334154.

a(n) = n! + sum(j=0, n-1, (j+1)*j!); \\ Michel Marcus, May 11 2020

a(n) = n! + Sum_{j=0..n-1} (j+1)*j!.

A334154 a(n) is the number of length n decorated permutations avoiding the pattern 012.

Original entry on oeis.org

1, 2, 5, 15, 54, 236, 1254, 7986, 59584, 509304, 4897272, 52237448, 611460432, 7787383488, 107155194928, 1583776282704, 25019083516416, 420609003810944, 7496930998018176, 141203784944996736, 2802115237399913728, 58432523737192745472, 1277372108617847278848

Offset: 0

Jordan Weaver, Apr 16 2020

nonn

A decorated permutation of length n is a word w=w_1...w_n on the letters {0,...,n} such that the restriction of w to its nonzero entries is an ordinary permutation in one-line notation. Then w avoids the pattern 012 if there is no subword w_{i_1}w_{i_2}w_{i_3} with i_1 < i_2 < i_3 such that w_{i_1} = 0 and 0 < w_{i_2} < w_{i_3}.

			For n=3, there are 16 decorated permutations of length 3 (000, 001, 010, 100, 012, 102, 120, 021, 201, 210, 123, 132, 213, 231, 312, and 321). All of these avoid 012 except 012 itself. Therefore, a(3) = 15.
For n=5, 02031 is a decorated permutation that does not avoid 012 because it contains the subword 023.

Cf. A334155, A334156.

a(n) = n! + sum(j=1, n, sum(l=1, n-j+1, binomial(n-l,j-1)*binomial(n-j,l-1)*(l-1)!)); \\ Michel Marcus, May 11 2020

a(n) = n! + Sum_{j=1..n} Sum_{l=1..n-j+1} binomial(n-l,j-1)*binomial(n-j,l-1)*(l-1)!.

A322481 Permutation breadth triangle: B(n,k) is the number of permutations w in S_n with breadth(w) = k, where breadth(w) = min({ |i-j|+|w(i)-w(j)| : 1 <= i < j <= n }).

Original entry on oeis.org

0, 0, 2, 0, 6, 0, 0, 22, 2, 0, 0, 106, 14, 0, 0, 0, 630, 90, 0, 0, 0, 0, 4394, 644, 2, 0, 0, 0, 0, 35078, 5222, 20, 0, 0, 0, 0, 0, 315258, 47464, 158, 0, 0, 0, 0, 0, 0, 3149494, 477346, 1960, 0, 0, 0, 0, 0, 0

Offset: 1

Jordan Weaver, Dec 10 2018

B(n,1) = 0 for all n, because for any 1<=i,j<=n and any w in S_n, 2 <= |i-j|+|w(i)-w(j)| <= breadth(w).

			For n=4, k=3, the B(4,3) = 2 permutations in S_4 with breadth 3 are [2,4,1,3] and [3,1,4,2] in one-line notation.
Triangle: B(n,k) begins:
  0;
  0,       2;
  0,       6,      0;
  0,      22,      2,    0;
  0,     106,     14,    0, 0;
  0,     630,     90,    0, 0, 0;
  0,    4394,    644,    2, 0, 0, 0;
  0,   35078,   5222,   20, 0, 0, 0, 0;
  0,  315258,  47464,  158, 0, 0, 0, 0, 0;
  0, 3149494, 477346, 1960, 0, 0, 0, 0, 0, 0;

D. Bevan, C. Homberger, and B. E. Tenner, Prolific permutations and permuted packings: downsets containing many large patterns, arXiv:1608.06931 [math.CO], 2016_2017; J. Combin. Theory A., 153:98-121, 2018.

Column k=2 gives A129535.

Row sums give A000142 (for n>1).

cp's OEIS Frontend

User: Jordan Weaver

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

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

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A334156 Triangle read by rows: T(n,m) is the number of length n decorated permutations avoiding the word 0^m = 0...0 of m 0's, where 1 <= m <= n.

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A334155 a(n) is the number of length n decorated permutations avoiding the pattern 001.

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A334154 a(n) is the number of length n decorated permutations avoiding the pattern 012.

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