A154655 -id:A154655 - cp's OEIS frontend

A002524 Number of permutations of length n within distance 2 of a fixed permutation.

1, 1, 2, 6, 14, 31, 73, 172, 400, 932, 2177, 5081, 11854, 27662, 64554, 150639, 351521, 820296, 1914208, 4466904, 10423761, 24324417, 56762346, 132458006, 309097942, 721296815, 1683185225, 3927803988, 9165743600, 21388759708, 49911830577, 116471963129

Offset: 0

N. J. A. Sloane

From Torleiv Kløve, Jan 09 2009: (Start)
Let V(d,n) be the number of permutations of length n within distance d of a fixed permutation. For d=1,2,3,4,...,10 these are A000045, A002524, A002526, A072856, A154654, A154655, A154656, A154657, A154658, A154659. The generating function is a rational function f_d(z)/g_d(z) (see the Kløve report referenced here). For d<=6, deg g_d = 2^{d-1} + binomial(2*d,d)/2 and deg f_d(z) = deg g_d(z)-2d. As a table:
   d deg g_d deg f_d
   1    2       0
   2    5       1
   3   14       8
   4   43      35
   5  142     132
   6  494     482
(End)
For positive n, a(n) equals the permanent of the n X n matrix with 1's along the five central diagonals, and 0's everywhere else. - John M. Campbell, Jul 09 2011

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics I, Example 4.7.16, p. 253.

R. H. Hardin, Table of n, a(n) for n = 0..400, Jul 11 2010
V. Baltic, On the number of certain types of strongly restricted permutations, Appl. An. Disc. Math. 4 (2010), 119-135; DOI:10.2298/AADM1000008B.
M. Barnabei, F. Bonetti and M. Silimbani, Two permutation classes related to the Bubble Sort operator, Electronic Journal of Combinatorics 19(3) (2012), #P25. - From _N. J. A. Sloane_, Dec 25 2012
Fan R. K. Chung, Persi Diaconis, and Ron Graham, Permanental generating functions and sequential importance sampling, Stanford University (2018).
Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement, Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.
Torleiv Kløve, Generating functions for the number of permutations with limited displacement, Electron. J. Combin., 16 (2009), #R104. - From _N. J. A. Sloane_, May 04 2011.
O. Krafft and M. Schaefer, On the number of permutations within a given distance, Fib. Quart. 40 (5) (2002) 429-434.
Ting Kuo, K-sorted Permutations with Weakly Restricted Displacements, Journal of Computers, 2014. See p. 36.
R. Lagrange, Quelques résultats dans la métrique des permutations, Annales Scientifiques de l'École Normale Supérieure, Paris, 79 (1962), 199-241.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Andrew Tsao, Sampling Methodology for Intractable Counting Problems, Ph. D. Thesis, Stanford University (2020).
Index entries for linear recurrences with constant coefficients, signature (2,0,2,0,-1).

Column k=2 of A306209.

I:=[1,1,2,6,14]; [n le 5 select I[n] else 2*Self(n-1) +2*Self(n-3) -Self(n-5): n in [1..41]]; // G. C. Greubel, Jan 21 2022

CoefficientList[Series[(1-x)/(1-2*x-2*x^3+x^5), {x,0,50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 24 2011 *)

a(n)=if(n,([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,0,2,0,2]^n*[1;1;2;6;14])[1,1],1) \\ Charles R Greathouse IV, Jul 28 2015

[( (1-x)/(1-2*x-2*x^3+x^5) ).series(x,n+1).list()[n] for n in (0..40)] # G. C. Greubel, Jan 21 2022

G.f.: (1-x)/(1-2*x-2*x^3+x^5). - Simon Plouffe in his 1992 dissertation.

Typo in comment corrected by Vaclav Kotesovec, Dec 01 2012

A306209 Number A(n,k) of permutations of [n] within distance k of a fixed permutation; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 6, 5, 1, 1, 1, 2, 6, 14, 8, 1, 1, 1, 2, 6, 24, 31, 13, 1, 1, 1, 2, 6, 24, 78, 73, 21, 1, 1, 1, 2, 6, 24, 120, 230, 172, 34, 1, 1, 1, 2, 6, 24, 120, 504, 675, 400, 55, 1, 1, 1, 2, 6, 24, 120, 720, 1902, 2069, 932, 89, 1, 1, 1, 2, 6, 24, 120, 720, 3720, 6902, 6404, 2177, 144, 1

Offset: 0

Alois P. Heinz, Jan 29 2019

A(n,k) counts permutations p of [n] such that |p(j)-j| <= k for all j in [n].

			A(4,1) = 5: 1234, 1243, 1324, 2134, 2143.
A(5,2) = 31: 12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13524, 14235, 14253, 14325, 14523, 21345, 21354, 21435, 21453, 21534, 21543, 23145, 23154, 24135, 24153, 31245, 31254, 31425, 31524, 32145, 32154, 34125.
Square array A(n,k) begins:
  1,  1,   1,    1,    1,     1,     1,     1,     1, ...
  1,  1,   1,    1,    1,     1,     1,     1,     1, ...
  1,  2,   2,    2,    2,     2,     2,     2,     2, ...
  1,  3,   6,    6,    6,     6,     6,     6,     6, ...
  1,  5,  14,   24,   24,    24,    24,    24,    24, ...
  1,  8,  31,   78,  120,   120,   120,   120,   120, ...
  1, 13,  73,  230,  504,   720,   720,   720,   720, ...
  1, 21, 172,  675, 1902,  3720,  5040,  5040,  5040, ...
  1, 34, 400, 2069, 6902, 17304, 30960, 40320, 40320, ...

Alois P. Heinz, Antidiagonals n = 0..36, flattened
Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement, Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.
Torleiv Kløve, Generating functions for the number of permutations with limited displacement, Electron. J. Combin., 16 (2009), #R104.

Columns k=0..10 give: A000012, A000045(n+1), A002524, A002526, A072856, A154654, A154655, A154656, A154657, A154658, A154659.
Rows n=1-2 give: A000012, A040000.
Main diagonal gives A000142.
A(2n,n) gives A048163(n+1).
A(2n+1,n) gives A092552(n+1).
A(n,floor(n/2)) gives A306267.
A(n+2,n) gives A001564.
Cf. A130152.

A[0, _] = 1;
A[n_ /; n > 0, k_] := A[n, k] = Permanent[Table[If[Abs[i - j] <= k, 1, 0], {i, 1, n}, {j, 1, n}]];
Table[A[n - k, k], {n, 0, 12}, {k, n, 0, -1 }] // Flatten (* Jean-François Alcover, Oct 18 2021, after Alois P. Heinz in A130152 *)

A(n,k) = Sum_{j=0..k} A130152(n,j) for n > 0, A(0,k) = 1.

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

A002524 Number of permutations of length n within distance 2 of a fixed permutation.

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A306209 Number A(n,k) of permutations of [n] within distance k of a fixed permutation; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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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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A323802 Number of permutations p of [n] such that max_{j=1..n} |p(j)-j| = 6.

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A323803 Number of permutations p of [n] such that max_{j=1..n} |p(j)-j| = 7.

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