cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A338838 Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] where adjacent values cannot be consecutive modulo n.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 0, 0, 1, 4, 4, 0, 0, 1, 5, 10, 10, 10, 10, 1, 6, 18, 36, 60, 84, 60, 1, 7, 28, 84, 210, 434, 630, 462, 1, 8, 40, 160, 544, 1552, 3440, 5168, 3920, 1, 9, 54, 270, 1170, 4338, 13158, 30366, 47178, 36954, 1, 10, 70, 420, 2220, 10220, 39780, 125220, 298060, 476220, 382740
Offset: 0

Views

Author

Xiangyu Chen, Nov 11 2020

Keywords

Comments

In a convex n-gon, the number of paths using k-1 diagonals and k non-repeated vertices.

Examples

			n\k  0    1    2    3    4    5    6    7    8
0    1
1    1    1
2    1    2    0
3    1    3    0    0
4    1    4    4    0    0
5    1    5    10   10   10   10
6    1    6    18   36   60   84   60
7    1    7    28   84   210  434  630  462
8    1    8    40   160  544  1552 3440 5168 3920

Crossrefs

Right diagonal is A002493.
Cf. A011973, A034807, A338526, A338849, A000166.

Programs

  • PARI
    isokd(d, n) = my(x=abs(d)); (x==1) || (x==(n-1));
isok(s, p, n) = {my(w = vector(#s, k, s[p[k]])); for (i=1, #s-1, if (isokd(w[i+1] - w[i], n) == 1, return (0))); return (1);}
T(n, k) = {my(nb = 0); forsubset([n, k], s, for(i=1, k!, if (isok(s, numtoperm(k, i), n), nb++););); nb;} \\ Michel Marcus, Nov 21 2020

Formula

T(n,k) = n*(A338526(n-1,k-1)-S(n-1,k-1)) for k>0 except T(2,2)=0, T(n,0)=1, where S(n,k) = 2*A338526(n-1,k-1)-S(n-1,k-1) for k>0, S(n,0)=0.

A338849 Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] in a circle where adjacent values cannot be consecutive modulo n, rotations are distinct.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 0, 0, 1, 4, 4, 0, 0, 1, 5, 10, 0, 0, 10, 1, 6, 18, 12, 24, 60, 36, 1, 7, 28, 42, 112, 280, 420, 322, 1, 8, 40, 96, 336, 1040, 2400, 3696, 2832, 1, 9, 54, 180, 792, 3060, 9540, 22428, 35280, 27954, 1, 10, 70, 300, 1600, 7540, 29880, 95340, 229280, 369540, 299260
Offset: 0

Views

Author

Xiangyu Chen, Nov 12 2020

Keywords

Comments

In a convex n-gon, the number of paths using k-1 diagonals and k non-repeated vertices, start and end vertices are not connected by a side.

Examples

			n\k  0    1    2    3    4    5    6    7    8
0    1
1    1    1
2    1    2    0
3    1    3    0    0
4    1    4    4    0    0
5    1    5    10   0    0    10
6    1    6    18   12   24   60   36
7    1    7    28   42   112  280  420  322
8    1    8    40   96   336  1040 2400 3696 2832

Crossrefs

Cf. A011973, A034807, A338526, A338838, A000166.

Formula

T(n,k) = n*(A338526(n-1,k-1)-2*S(n-1,k-1)+Z2(n-1,k-1)) for k>0 except T(2,2)=0, T(n,0)=1, where Z2(n,k) = Z1(n,k) except Z2(n,1)=2, where Z1(n,k) = S(n-1,k-1)-Z(n-1,k-1) for k>0 except Z1(2,2)=0, Z1(n,0)=0, where S(n,k) = 2*A338526(n-1,k-1)-S(n-1,k-1) for k>0, S(n,0)=0.

A340106 Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] with longest consecutive chain size less than 3.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 4, 1, 4, 12, 20, 18, 1, 5, 20, 54, 100, 92, 1, 6, 30, 112, 318, 600, 570, 1, 7, 42, 200, 768, 2208, 4244, 4082, 1, 8, 56, 324, 1570, 6080, 17682, 34300, 33292, 1, 9, 72, 490, 2868, 13980, 54552, 159702, 311808, 304490, 1, 10, 90, 704, 4830, 28392, 139130, 545528, 1604616, 3147164, 3086890
Offset: 0

Views

Author

Xiangyu Chen, Dec 28 2020

Keywords

Comments

In a convex n-gon, the number of paths using k non-repeated vertices and fewer than 3 vertices (2 sides) in a row, except side (1,n) is unrestricted.

Examples

			n\k   0     1      2      3      4     5     6     7     8
0     1
1     1     1
2     1     2      2
3     1     3      6      4
4     1     4     12     20     18
5     1     5     20     54    100    92
6     1     6     30    112    318   600    570
7     1     7     42    200    768  2208   4244  4082
8     1     8     56    324   1570  6080  17682 34300 33292

Crossrefs

Right diagonal is A095816.
Cf. A338526, A338838, A338849.
Cf. A340107, A340108.

Formula

T(n,k) = A340107(n,k) + 2*O(n-1,k-1) + O(n-2,k-2), where O(n,k) = 2*(k-1)*T(n-1,k-1)/(n-1) - 2*O(n-1,k-1) + 3*O(n-2,k-2) + 2*O(n-3,k-3) + O(n-4,k-4), O(n,k)=0 for k<=1.

A340107 Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] with longest consecutive chain size less than 3, when 1 and n are considered to be consecutive.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 0, 1, 4, 12, 16, 16, 1, 5, 20, 50, 90, 80, 1, 6, 30, 108, 300, 552, 516, 1, 7, 42, 196, 742, 2100, 3990, 3794, 1, 8, 56, 320, 1536, 5888, 16976, 32656, 31456, 1, 9, 72, 486, 2826, 13680, 53046, 154350, 299628, 290970, 1, 10, 90, 700, 4780, 27960, 136380, 532340, 1559040, 3044900, 2974380
Offset: 0

Views

Author

Xiangyu Chen, Dec 28 2020

Keywords

Comments

In a convex n-gon, the number of paths using k non-repeated vertices and fewer than 3 vertices (2 sides) in a row.

Examples

			n\k   0     1     2     3     4     5     6     7     8
0     1
1     1     1
2     1     2      2
3     1     3      6      0
4     1     4     12     16     16
5     1     5     20     50     90    80
6     1     6     30    108    300   552    516
7     1     7     42    196    742  2100   3990  3794
8     1     8     56    320   1536  5888  16976 32656 31456

Crossrefs

Cf. A338526, A338838, A338849.
Cf. A340106, A340108.

Formula

T(n,k) = n*(A340106(n-1,k-1) - S(n-2,k-2)) except for T(n,0)=1, where S(n,k) = 2*A340106(n-1,k-1) - 2*A340106(n-2,k-2) + S(n-3,k-3), S(n,k)=0 for k <= 0. [exception added by Xiangyu Chen, Aug 19 2022]

A340108 Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] in a circle with longest consecutive chain size less than 3, when 1 and n are considered to be consecutive, and rotations are considered to be distinct.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 0, 1, 4, 12, 0, 16, 1, 5, 20, 30, 80, 60, 1, 6, 30, 84, 264, 480, 456, 1, 7, 42, 168, 672, 1890, 3612, 3458, 1, 8, 56, 288, 1424, 5440, 15744, 30352, 29296, 1, 9, 72, 450, 2664, 12870, 50004, 145656, 283104, 275166, 1, 10, 90, 660, 4560, 26640, 130080, 508060, 1488960, 2909700, 2843980
Offset: 0

Views

Author

Xiangyu Chen, Dec 28 2020

Keywords

Comments

In a convex n-gon, the number of cycles using k non-repeated vertices and fewer than 3 vertices (2 sides) in a row.

Examples

			n\k   0     1      2      3     4     5     6     7     8
0     1
1     1     1
2     1     2      2
3     1     3      6      0
4     1     4     12      0     16
5     1     5     20     30     80    60
6     1     6     30     84    264   480    456
7     1     7     42    168    672  1890   3612  3458
8     1     8     56    288   1424  5440  15744 30352 29296

Crossrefs

Cf. A338526, A338838, A338849.
Cf. A340106, A340107.

Formula

T(n,k) = n*(5*A340106(n-1,k-1) - 2*Z(n,k) - Z(n-1,k-1) - 2*S(n,k) - 2*S(n-2,k-2)) except for T(n,0)=1, where S(n,k) = 2*A340106(n-1,k-1) - 2*A340106(n-2,k-2) + S(n-3,k-3), S(n,k)=0 for k <= 0, Z(n,k) = 2*A340106(n-1,k-1) - S(n,k) - V(n-1,k-1), Z(n,k)=0 for k <= 0, V(n,k) = Z(n-1,k-1) - V(n-1,k-1), V(n,k)=0 for k <= 0 except for V(2,2)=2.

Extensions

Terms of column 2 corrected by Xiangyu Chen, Aug 19 2022

A375022 Number of permutations of n (distinct) elements from [2n] without consecutive adjacent values.

Original entry on oeis.org

1, 2, 6, 48, 650, 11604, 253890, 6572176, 196467642, 6660672060, 252504829346, 10584157180920, 486052773276954, 24267623565599428, 1308816525199710210, 75828568345500497184, 4696852810090223180090, 309727816544792336406156, 21664419864179709448539042
Offset: 0

Views

Author

Alois P. Heinz, Aug 04 2024

Keywords

Examples

			a(0) = 1: the empty permutation.
a(1) = 2: 1, 2.
a(2) = 6: 13, 14, 24, 31, 41, 42.
a(3) = 48: 135, 136, 142, 146, 152, 153, 162, 163, 164, 241, 246, 251, 253, 261, 263, 264, 314, 315, 316, 351, 352, 361, 362, 364, 413, 415, 416, 425, 426, 461, 462, 463, 513, 514, 516, 524, 526, 531, 536, 613, 614, 615, 624, 625, 631, 635, 641, 642.

Links

Crossrefs

Cf. A338526.

Formula

a(n) = A338526(2n,n).
Showing 1-6 of 6 results.

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