A080013 -id:A080013 - cp's OEIS frontend

A264550 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change -1,1 -1,2 1,0 or 0,-1.

0, 1, 1, 1, 2, 0, 1, 6, 5, 0, 1, 16, 16, 10, 1, 3, 40, 36, 45, 21, 0, 3, 96, 172, 216, 133, 44, 0, 4, 240, 764, 1528, 1160, 400, 93, 1, 6, 608, 2728, 11728, 14852, 4640, 1204, 196, 0, 9, 1536, 9880, 66372, 163744, 105081, 23140, 3561, 413, 0, 12, 3840, 38818, 403920

Offset: 1

R. H. Hardin, Nov 17 2015

Table starts
.0...1.....1.......1.........1...........3.............3...............4
.1...2.....6......16........40..........96...........240.............608
.0...5....16......36.......172.........764..........2728............9880
.0..10....45.....216......1528.......11728.........66372..........403920
.1..21...133....1160.....14852......163744.......1573479........16103502
.0..44...400....4640....105081.....1994830......29691127.......480994368
.0..93..1204...23140....813473....25172836.....591482528.....15554895920
.1.196..3561..110592...6587227...316806958...11959837149....510948245264
.0.413.10554..501828..50597392..3939510885..234588752099..16034698666868
.0.870.31493.2390752.392445418.49105816488.4630258704238.508458281055496

			Some solutions for n=4 k=4
..1..2..5..6..7....1..2..3..4..7....1..2..3..7..8....1..5..3..6..8
..0.10..8..3..4....0.10..8.11.12....0.10.11.12..4....0..7..2.12..4
.11.12.15.14..9....5..6.13.17..9....5..6.13.17..9...11.15.13.14..9
.16.20.18.13.23...16.20.18.21.14...16.20.18.21.14...10.20.18.22.23
.21.22.17.24.19...15.22.23.24.19...15.22.23.24.19...21.16.17.24.19

R. H. Hardin, Table of n, a(n) for n = 1..198

Row 1 is A080013(n+1).

Empirical for column k:
k=1: a(n) = a(n-3)
k=2: a(n) = 2*a(n-1) +a(n-4)
k=3: [order 15]
k=4: [order 10] for n>11
k=5: [order 84]
Empirical for row n:
n=1: a(n) = a(n-2) +a(n-3) +a(n-4) -a(n-6)
n=2: a(n) = 2*a(n-1) +8*a(n-4)
n=3: [order 70]
n=4: [order 56] for n>59

A264676 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change -1,-1 1,0 -1,-2 -2,-2 or 0,1.

Original entry on oeis.org

0, 1, 1, 1, 2, 0, 1, 10, 9, 0, 1, 29, 34, 19, 1, 3, 75, 123, 145, 44, 0, 3, 201, 748, 890, 603, 108, 0, 4, 588, 3698, 9205, 6851, 2417, 264, 1, 6, 1700, 17443, 88687, 123222, 43131, 9976, 649, 0, 9, 4785, 84737, 714235, 2025372, 1449467, 291315, 40825, 1573, 0, 12

Offset: 1

R. H. Hardin, Nov 20 2015

Table starts
.0....1......1........1...........1.............3..............3
.1....2.....10.......29..........75...........201............588
.0....9.....34......123.........748..........3698..........17443
.0...19....145......890........9205.........88687.........714235
.1...44....603.....6851......123222.......2025372.......29875350
.0..108...2417....43131.....1449467......43095017.....1095840260
.0..264...9976...291315....17406354.....918298857....40765995197
.1..649..40825..1980287...209749118...19297801298..1494347569074
.0.1573.166985.13199209..2503676777..404856272120.54204721906375
.0.3837.684253.88670692.29934965340.8489260394876

			Some solutions for n=4 k=4
..6..8..1..2..3....7..8..1..2..3...12..8..1..2..3...12.13..1..2..3
..0..5.14..7..4....0.13..6.14..4....0..5..6..7..4....0.18.19..7..4
.16.10.24.19..9....5.23.24.12..9...17.18.24.19..9....5..6.24..8..9
.21.11.12.13.18...10.11.16.17.18...10.11.16.13.14...10.11.16.17.14
.15.20.17.22.23...15.20.21.22.19...15.20.21.22.23...15.20.21.22.23

R. H. Hardin, Table of n, a(n) for n = 1..127

Row 1 is A080013(n+1).

Empirical for column k:
k=1: a(n) = a(n-3)
k=2: a(n) = 2*a(n-1) +a(n-3) +3*a(n-4) +2*a(n-5) -a(n-6) +4*a(n-7) -a(n-8) -a(n-10)
k=3: [order 15]
k=4: [order 26] for n>27
Empirical for row n:
n=1: a(n) = a(n-2) +a(n-3) +a(n-4) -a(n-6)
n=2: [order 9]
n=3: [order 70]
n=4: [order 81] for n>86

A264422 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change 0,1 2,2 1,0 -1,2 -2,-1 or -1,-1.

Original entry on oeis.org

0, 2, 1, 0, 3, 1, 0, 12, 20, 1, 4, 37, 96, 85, 1, 0, 119, 499, 529, 351, 3, 0, 385, 2681, 7311, 4843, 1462, 3, 8, 1252, 15088, 101862, 137584, 44264, 6021, 4, 0, 4061, 86469, 1103666, 4023076, 2486429, 352713, 25188, 6, 0, 13166, 479787, 13805196, 95946259

Offset: 1

R. H. Hardin, Nov 12 2015

Table starts
.0......2.........0...........0............4............0............0
.1......3........12..........37..........119..........385.........1252
.1.....20........96.........499.........2681........15088........86469
.1.....85.......529........7311.......101862......1103666.....13805196
.1....351......4843......137584......4023076.....95946259...2486418295
.3...1462.....44264.....2486429....147984724...7577203142.396849643846
.3...6021....352713....40217261...5498758183.577598946184
.4..25188...2840274...681146169.201733617932
.6.104870..23795758.11846231201
.9.437164.197390196

			Some solutions for n=4 k=4
..6..7..8..2..3....6..7..8..2..3....6..0..8..9..3....6.12.13.14..3
..0..1.13.19..4...11..1.13.19..4...16..1..2..7..4....0..1..2.19..4
..5.17.18.12..9....5.17..0.12..9....5.17.11.19.13...16.10..7..8..9
.21.22.20.24.14...10.22.16.24.14...10.22.23.24.14...21.11..5.24.22
.15.16.10.11.23...15.20.21.18.23...15.20.21.18.12...15.20.17.18.23

R. H. Hardin, Table of n, a(n) for n = 1..84

Column 1 is A080013(n+1).

Empirical for column k:
k=1: a(n) = a(n-2) +a(n-3) +a(n-4) -a(n-6)
k=2: [order 56]
k=3: [order 70]
Empirical for row n:
n=1: a(n) = 2*a(n-3)
n=2: a(n) = 3*a(n-1) +a(n-3) +5*a(n-4) +a(n-7)
n=3: [order 33]
n=4: [order 16] for n>18

A376743 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,-1,4} for all i=1,...,n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 5, 5, 6, 8, 11, 15, 25, 35, 46, 61, 85, 125, 175, 245, 341, 470, 650, 925, 1300, 1810, 2521, 3520, 4915, 6880, 9640, 13476, 18801, 26251, 36721, 51346, 71776, 100335, 140210, 195886, 273813, 382821, 535105, 747850, 1045220

Offset: 0

Michael A. Allen, Oct 03 2024

Other sequences related to strongly restricted permutations pi(i) of i in {1,..,n} along with the sets of allowed p(i)-i (containing at least 3 elements): A000045 {-1,0,1}, A189593 {-1,0,2,3,4,5,6}, A189600 {-1,0,2,3,4,5,6,7}, A006498 {-2,0,2}, A080013 {-2,1,2}, A080014 {-2,0,1,2}, A033305 {-2,-1,1,2}, A002524 {-2,-1,0,1,2}, A080000 {-2,0,3}, A080001 {-2,1,3}, A080004 {-2,0,1,3}, A080002 {-2,2,3}, A080005 {-2,0,2,3}, A080008 {-2,1,2,3}, A080011 {-2,0,1,2,3}, A079999 {-2,-1,3}, A080003 {-2,-1,0,3}, A080006 {-2,-1,1,3}, A080009 {-2,-1,0,1,3}, A080007 {-2,-1,2,3}, A080010 {-2,-1,0,2,3}, A080012 {-2,-1,1,2,3}, A072827 {-2,-1,0,1,2,3}, A224809 {-2,0,4}, A189585 {-2,0,1,3,4}, A189581 {-2,-1,0,3,4}, A072850 {-2,-1,0,1,2,3,4}, A189587 {-2,0,1,3,4,5}, A189588 {-2,-1,0,3,4,5}, A189594 {-2,0,1,3,4,5,6}, A189595 {-2,-1,0,3,4,5,6}, A189601 {-2,0,1,3,4,5,6,7}, A189602 {-2,-1,0,3,4,5,6,7}, A224811 {-2,0,8}, A224812 {-2,0,10}, A224813 {-2,0,12}, A006500 {-3,0,3}, A079981 {-3,1,3}, A079983 {-3,0,1,3}, A079982 {-3,2,3}, A079984 {-3,0,2,3}, A079988 {-3,1,2,3}, A079989 {-3,0,1,2,3}, A079986 {-3,-1,1,3}, A079992 {-3,-1,0,1,3}, A079987 {-3,-1,2,3}, A079990 {-3,-1,0,2,3}, A079993 {-3,-1,1,2,3}, A079985 {-3,-2,2,3}, A079991 {-3,-2,0,2,3}, A079996 {-3,-2,0,1,2,3}, A079994 {-3,-2,1,2,3}, A079997 {-3,-2, -1,1,2,3}, A002526 {-3,-2,-1,0,1,2,3}, A189586 {-3,0,1,2,4}, A189583 {-3,-1,0,2,4}, A189582 {-3,-2,0,1,4}, A189584 {-3,-2,-1,0,4}, A189589 {-3,0,1,2,4,5}, A189590 {-3,-1,0,2,4,5}, A189591 {-3,-2,1,4,5}, A189592 {-3,-2,-1,0,4,5}, A224810 {-3,0,6}, A189596 {-3,0,1,2,4,5,6}, A189597 {-3,-1,0,2,4,5,6}, A189598 {-3,-2,0,1,4,5,6}, A189599 {-3,-2,-1,0,4,5,6}, A224814 {-3,0,9}, A031923 {-4,0,4}, A072856 {-4,-3, -2,-1,0,1,2,3,4}, A224815 {-4,0,8}, A154654 {-5,-4,-3,-2,-1,0,1,2,3,4,5}, A154655 {-6,-5,-4,-3, -2,-1,0,1,2,3,4,5,6}.

[Keyword "less", because this comment should be moved to the Index to the OEIS, it is not appropriate here. - N. J. A. Sloane, Oct 25 2024]

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), North-Holland, Amsterdam, 1970, pp. 755-770.

Michael A. Allen and Kenneth Edwards, Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices, Lin. Multilin. Alg. 72:13 (2024) 2091-2103.
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics, 4(1) (2010), 119-135.
Kenneth Edwards and Michael A. Allen, Strongly restricted permutations and tiling with fences, Discrete Applied Mathematics, 187 (2015), 82-90.
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,1,2,1,0,-2,-2,0,-1,0,0,1).

See comments for other sequences related to strongly restricted permutations.

CoefficientList[Series[(1 - x^3 - x^4 - x^6 + x^9)/(1 - x^3 - x^4 - x^5 - 2*x^6 - x^7 + 2*x^9 + 2*x^10 + x^12 - x^15),{x,0,49}],x]
LinearRecurrence[{0, 0, 1, 1, 1, 2, 1, 0, -2, -2, 0, -1, 0, 0, 1}, {1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 5, 5, 6, 8}, 50]

a(n) = a(n-3) + a(n-4) + a(n-5) + 2*a(n-6) + a(n-7) - 2*a(n-9) - 2*a(n-10) - a(n-12) + a(n-15).

G.f.: (1 - x^3 - x^4 - x^6 + x^9)/(1 - x^3 - x^4 - x^5 - 2*x^6 - x^7 + 2*x^9 + 2*x^10 + x^12 - x^15).

cp's OEIS Frontend

A264550 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change -1,1 -1,2 1,0 or 0,-1.

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A264676 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change -1,-1 1,0 -1,-2 -2,-2 or 0,1.

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Formula

A264422 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change 0,1 2,2 1,0 -1,2 -2,-1 or -1,-1.

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Crossrefs

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A376743 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,-1,4} for all i=1,...,n.

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