A306209 -id:A306209 - 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

A072856 Number of permutations satisfying i-4<=p(i)<=i+4, i=1..n (permutations of length n within distance 4).

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 504, 1902, 6902, 25231, 95401, 365116, 1396948, 5316192, 20135712, 76227216, 288878956, 1095937420, 4159450913, 15783649241, 59878012558, 227128287882, 861543171080, 3268198646496, 12398132725784, 47033439463906, 178423731589482

Offset: 0

Vladimir Baltic, Jul 25 2002

a(n) equals the permanent of the n X n matrix with 1's along the nine central diagonals and 0's everywhere else. - John M. Campbell, Jul 09 2011

R. H. Hardin, Table of n, a(n) for n = 0..400 (corrected by _R. H. Hardin_, Jan 19 2019)
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135
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.
Index entries for linear recurrences with constant coefficients, signature (3, 2, -1, -1, 70, 39, -31, -114, -522, -184, 34, 46, 1444, 202, -606, 1204, 198, -804, 542, -26, -2372, -318, 1582, -328, -2018, -222, 810, 184, -706, -14, 204, 70, 14, 28, -22, -11, 47, -8, -11, -1, -4, 1, 1).

Cf. A002524..A002529, A072827, A072850..A072856, A079955..A080014.

Column k=4 of A306209.

G.f.: (1 -2*x -3*x^2 -x^3 +4*x^4 -31*x^5 -5*x^6 +32*x^7 -21*x^8 +129*x^9 +94*x^10 -83*x^11 +11*x^12 -192*x^13 -59*x^14 +63*x^15 -16*x^16 +3*x^17 -29*x^18 -46*x^19 -57*x^20 +253*x^21 -28*x^22 -101*x^23 +17*x^24 +104*x^25 -15*x^26 -29*x^27 +10*x^28 -x^29 +x^30 -x^32 -3*x^33 +x^35) / (1 -3*x -2*x^2 +x^3 +x^4 -70*x^5 -39*x^6 +31*x^7 +114*x^8 +522*x^9 +184*x^10 -34*x^11 -46*x^12 -1444*x^13 -202*x^14 +606*x^15 -1204*x^16 -198*x^17 +804*x^18 -542*x^19 +26*x^20 +2372*x^21 +318*x^22 -1582*x^23 +328*x^24 +2018*x^25 +222*x^26 -810*x^27 -184*x^28 +706*x^29 +14*x^30 -204*x^31 -70*x^32 -14*x^33 -28*x^34 +22*x^35 +11*x^36 -47*x^37 +8*x^38 +11*x^39 +x^40 +4*x^41 -x^42 -x^43). - Torleiv Kløve, Jan 13 2009; corrected by Colin Barker, Jul 06 2013

a(0)=1 prepended and more terms added by Colin Barker, Jul 06 2013

A001564 2nd differences of factorial numbers.

Original entry on oeis.org

1, 3, 14, 78, 504, 3720, 30960, 287280, 2943360, 33022080, 402796800, 5308934400, 75203251200, 1139544806400, 18394619443200, 315149522688000, 5711921639424000, 109196040425472000, 2196014181064704000, 46346783255764992000, 1024251745442365440000

Offset: 0

N. J. A. Sloane

a(n) is also the number of isolated fixed points (i.e. adjacent fixed points are not isolated) in all permutations of [n+2]. Example: a(2)=14 because we have (the isolated fixed points are marked) 1'423, 1'324', 1'342, 1'43'2, 413'2, 3124', 42'13, 2314', 243'1, 32'14', 32'41. - Emeric Deutsch, Apr 18 2009
The average of the first n terms is n factorial. - Franklin T. Adams-Watters, May 20 2010
Number of blocks in all permutations of [n+1]. A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. For example, the permutation 5412367 has 4 blocks: 5, 4, 123, and 67. Example: a(2)=14 because the permutations of [3], separated into blocks, are 123, 1-3-2, 2-1-3, 23-1, 3-12, 3-2-1 with 1+3+3+2+2+3=14 blocks. - Emeric Deutsch, Jul 12 2010
a(n) equals n+1 times the permanent of the (n+1) X (n+1) matrix with 1/(n+1) in the top right corner and 1's everywhere else. - John M. Campbell, May 25 2011
Number of permutations s of [n+2] where 2 designated elements are not mapped to themselves, e.g., s(1) != 1 and s(2) != 2. See Janjić article. - Benjamin Schreyer, May 07 2025

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).

T. D. Noe, Table of n, a(n) for n = 0..100
A. van Heemert, Cyclic permutations with sequences and related problems, J. Reine Angew. Math., 198 (1957), 56-72.
Milan Janjić, Enumerative Formulae for Some Functions on Finite Sets
A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, A 99 (2002), 345-357.
Index entries for sequences related to factorial numbers

Cf. A000142, A001563, A002061, A010027, A047920, A306209.

[(n^2+n+1)*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Apr 10 2015

seq(factorial(n)*(n^2+n+1), n = 0 .. 20); # Emeric Deutsch, Apr 18 2009

Range[0,20]! CoefficientList[Series[(1+x^2)/(1-x)^3,{x,0,20}],x]
Differences[Range[0, 25]!, 2] (* Paolo Xausa, Jul 17 2025 *)

Vec(serlaplace((1+x^2)/(1-x)^3 + O(x^30))) \\ Michel Marcus, Apr 10 2015

a(n) = (n^2 + n + 1)*n! = A002061(n-1)*A000142(n). - Mitch Harris, Jul 10 2008
E.g.f.: (1+x^2)/(1-x)^3.
a(n) = A001563(n+1) - A001563(n). - Robert Israel, Apr 13 2015
a(n) = A306209(n+2,n). - Alois P. Heinz, Jan 29 2019
D-finite with recurrence a(n) +(-n-3)*a(n-1) +(n-1)*a(n-2)=0. - R. J. Mathar, Jul 01 2022

Comment edited by Franklin T. Adams-Watters, May 20 2010

A002526 Number of permutations of length n within distance 3 of a fixed permutation.

Original entry on oeis.org

1, 1, 2, 6, 24, 78, 230, 675, 2069, 6404, 19708, 60216, 183988, 563172, 1725349, 5284109, 16177694, 49526506, 151635752, 464286962, 1421566698, 4352505527, 13326304313, 40802053896, 124926806216, 382497958000, 1171122069784, 3585709284968, 10978628154457

Offset: 0

N. J. A. Sloane

For positive n, a(n) equals the permanent of the n X n matrix with 1's along the seven 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. 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.
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. (Table 3, top row).
O. Krafft and M. Schaefer, On the number of permutations within a given distance, Fib. Quart. 40 (5) (2002) 429-434.
R. Lagrange, Quelques résultats dans la métrique des permutations, Annales Scientifiques de l'École Normale Supérieure, Paris, 79 (1962), 199-241.
Index entries for linear recurrences with constant coefficients, signature (2,2,0,10,8,-2,-16,-10,-2,4,2,0,2,1).

The 14 sequences in Kløve's Table 3 are A002526, A002527, A002529, A188379, A188491, A188492, A188493, A188494, A002528, A188495, A188496, A188497, A188498, A002526.
Cf. A002524.
Column k=3 of A306209.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x-2*x^2-2*x^4+x^7+x^8)/(1-2*x-2*x^2-10*x^4-8*x^5+2*x^6+16*x^7+10*x^8 +2*x^9-4*x^10-2*x^11-2*x^13-x^14) )); // G. C. Greubel, Jan 22 2022

CoefficientList[Series[(1-x-2x^2-2x^4+x^7+x^8)/(1-2x-2x^2-10x^4-8x^5+ 2x^6+ 16x^7+10x^8+2x^9-4x^10-2x^11-2x^13-x^14),{x,0,50}],x] (* or *) LinearRecurrence[{2,2,0,10,8,-2,-16,-10,-2,4,2,0,2,1},{1,1,2,6,24,78, 230, 675,2069,6404,19708,60216,183988,563172},51] (* Harvey P. Dale, Jun 22 2011 *)

Vec((1-x-2*x^2-2*x^4+x^7+x^8)/(1-2*x-2*x^2-10*x^4-8*x^5+2*x^6+16*x^7+10*x^8+2*x^9-4*x^10-2*x^11-2*x^13-x^14)+O(x^99)) \\ Charles R Greathouse IV, Jul 16 2011

[( (1-x-2*x^2-2*x^4+x^7+x^8)/(1-2*x-2*x^2-10*x^4-8*x^5+2*x^6+16*x^7+10*x^8 +2*x^9-4*x^10-2*x^11-2*x^13-x^14) ).series(x,n+1).list()[n] for n in (0..40)] # G. C. Greubel, Jan 22 2022

G.f.: (1-x-2*x^2-2*x^4+x^7+x^8)/(1-2*x-2*x^2-10*x^4-8*x^5+2*x^6+16*x^7+10*x^8 +2*x^9-4*x^10-2*x^11-2*x^13-x^14).

a(0)=1, a(1)=1, a(2)=2, a(3)=6, a(4)=24, a(5)=78, a(6)=230, a(7)=675, a(8)=2069, a(9)=6404, a(10)=19708, a(11)=60216, a(12)=183988, a(13)=563172, a(n) = 2*a(n-1) +2*a(n-2) +10*a(n-4) +8*a(n-5) -2*a(n-6) -16*a(n-7) -10*a(n-8) -2*a(n-9) +4*a(n-10) +2*a(n-11) +2*a(n-13) +a(n-14). - Harvey P. Dale, Jun 22 2011

A092552 Let X_{m,n}(q) be the chromatic polynomial of the complete bipartite graph K_{m,n}. Then a(n) is the negative of the coefficient of the linear term of X_{n,n}(q).

Original entry on oeis.org

0, 1, 3, 31, 675, 25231, 1441923, 116914351, 12764590275, 1805409270031, 321113303226243, 70146437009397871, 18462286083671614275, 5762225835975165678031, 2104263061425865873128963, 888881838896989670838028591, 430058409024841744606172532675

Offset: 0

Michael Lugo (mtlugo(AT)mit.edu), Apr 09 2004

nonn

From Arvind Ayyer, Oct 25 2020: (Start)
Equivalently, a(n) is the number of acyclic orientations with a unique sink in K_{n,n}.
a(n) is also the number of toppleable permutations in S_{2n-1}. A toppleable permutation pi in S_{2n-1} satisfies pi_i <= n-1+i for 1 <= i <= n-1 and pi_i >= i-n+1 for n <= i <= 2n-1. The a(3)=3 toppleable permutations in S_3 are 123, 213 and 132. (End)

			a(2) = 3 since the chromatic polynomial of K_{2,2}(q) is q^4-4*q^3+6*q^2-3*q.
E.g.f.: A(x) = x + 3*x^2/2! + 31*x^3/3! + 675*x^4/4! + 25231*x^5/5! +...
where A(x) = (1-exp(-x)) + (1-exp(-2*x))^2/2 + (1-exp(-3*x))^3/3 +... - _Paul D. Hanna_, Dec 06 2012
O.g.f.: F(x) = x + 3*x^2 + 31*x^3 + 675*x^4 + 25231*x^5 +...
where F(x) = x/(1+x) + 2^1*2!*x^2/((1+2*1*x)*(1+2*2*x)) + 3^2*3!*x^3/((1+3*1*x)*(1+3*2*x)*(1+3*3*x)) + 4^3*4!*x^4/((1+4*1*x)*(1+4*2*x)*(1+4*3*x)*(1+4*4*x)) +... - _Paul D. Hanna_, Jan 05 2013

Alois P. Heinz, Table of n, a(n) for n = 0..238
A. Ayyer, D. Hathcock and P. Tetali, Toppleable Permutations, Excedances and Acyclic Orientations, arXiv:2010.11236 [math.CO], 2020. For the precise definition of a toppleable permutation.

Cf. A008277, A212084, A220179, A136126, A306209.

a:= n-> -coeff(add(Stirling2(n, k) *mul(q-i, i=0..k-1)
             *(q-k)^n, k=1..n), q, 1):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 30 2012

Table[Sum[k!*(k-1)!*StirlingS2[n,k]^2,{k,1,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 21 2013 *)

{a(n)=n!*polcoeff(sum(k=1,n,(1-exp(-k*x+x*O(x^n)))^k/k),n)} \\ Paul D. Hanna, Dec 06 2012
for(n=0,20,print1(a(n),", "))

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=if(n<=0, 0, sum(k=1, n, k!*(k-1)! * Stirling2(n, k)^2))} \\ Paul D. Hanna, Dec 30 2012
for(n=0, 20, print1(a(n), ", "))

{a(n)=if(n<1,0,polcoeff(sum(m=1, n, m^(m-1)*m!*x^m/prod(k=1, m, 1+m*k*x+x*O(x^n))), n))} \\ Paul D. Hanna, Jan 05 2013

From Alois P. Heinz, Apr 30 2012: (Start)
a(n) = (-1) * [q] Sum_{j=1..n} (q-j)^n*S2(n,j)*Product_{i=0..j-1} (q-i).
a(n) = (-1) * A212084(n,2n-1). (End)
E.g.f.: Sum_{n>=1} (1 - exp(-n*x))^n / n. - Paul D. Hanna, Dec 06 2012
a(n) = Sum_{k=1..n} k!*(k-1)! * Stirling2(n, k)^2. - Paul D. Hanna, Dec 30 2012, corrected by Vaclav Kotesovec, Jun 21 2013
O.g.f.: Sum_{n>=1} n^(n-1) * n! * x^n / Product_{k=1..n} (1 + n*k*x). - Paul D. Hanna, Jan 05 2013
a(n) = A136126(2*n-1,n), where triangle A136126(n,k) is the number of permutations of {1,2,...,k+n} having excedance set {1,2,...,k}. - Paul D. Hanna, Feb 01 2013
a(n) ~ sqrt(Pi) * n^(2*n-1/2) / (sqrt(1-log(2)) * exp(2*n) * (log(2))^(2*n)). - Vaclav Kotesovec, Nov 07 2014
a(n) = A306209(2n-1,n-1) for n > 0. - Alois P. Heinz, Feb 01 2019

A048163 a(n) = Sum_{k=1..n} ((k-1)!)^2*Stirling2(n,k)^2.

Original entry on oeis.org

1, 2, 14, 230, 6902, 329462, 22934774, 2193664790, 276054834902, 44222780245622, 8787513806478134, 2121181056663291350, 611373265185174628502, 207391326125004608457782, 81791647413265571604175094, 37109390748309009878392597910, 19192672725746588045912535407702

Offset: 1

N. J. A. Sloane, R. K. Guy

nonn

a(n) is also the number of max-closed relations on an ordered n-element domain (see the paper by Jeavons and Cooper, 1995). - Don Knuth, Feb 12 2024

			1
1 + 1 = 2
1 + 9 + 4 = 14
1 + 49 + 144 + 36 = 230
1 + 225 + 2500 + 3600 + 576 = 6902
... - _Philippe Deléham_, May 30 2015

Lovasz, L. and Vesztergombi, K.; Restricted permutations and Stirling numbers. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, pp. 731-738, Colloq. Math. Soc. Janos Bolyai, 18, North-Holland, Amsterdam-New York, 1978.
K. Vesztergombi, Permutations with restriction of middle strength, Stud. Sci. Math. Hungar., 9 (1974), 181-185.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Chad Brewbaker, A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues, INTEGERS Vol. 8 (2008), #A02.
Peter G. Jeavons and Martin C. Cooper, Tractable constraints on ordered domains, Artificial Intelligence 79 (1995), 327-339.
Hyeong-Kwan Ju and Seunghyun Seo, Enumeration of (0,1)-matrices avoiding some 2 X 2 matrices, Discrete Math., 312 (2012), 2473-2481.
Ken Kamano, Lonesum decomposable matrices, arXiv:1701.07157 [math.CO], 2017.
H.-K. Kim et al., Poly-Bernoulli numbers and lonesum matrices, arXiv:1103.4884 [math.CO], 2011.
Anatol N. Kirillov, On some quadratic algebras. I 1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 002, 172 p. (2016).

Main diagonal of array A099594.

Cf. A220181, A266695, A306209.

Table[Sum[((k-1)!)^2*StirlingS2[n,k]^2,{k,1,n}],{n,1,20}] (* Vaclav Kotesovec, Jun 21 2013 *)

a(n)=if(n<1, 0, polcoeff(sum(m=1, n, m^(m-1)*(m-1)!*x^m/prod(k=1, m-1, 1+m*k*x+x*O(x^n))), n)) \\ Paul D. Hanna, Jan 05 2013
for(n=1,20,print1(a(n),", "))

Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)
a(n)=sum(k=1,n,(-1)^(n-k)*k^(n-1)*(k-1)!*Stirling2(n-1, k-1))
for(n=1, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 06 2013

a(n) = sum(k=1, n, (k-1)!^2*stirling(n,k,2)^2); \\ Michel Marcus, Jun 22 2018

E.g.f. (with offset 0): Sum((1-exp(-(m+1)*z))^m, m=0..oo)
O.g.f.: Sum_{n>=1} n^(n-1) * (n-1)! * x^n / Product_{k=1..n-1} (1 - n*k*x). - Paul D. Hanna, Jan 05 2013
Limit n->infinity (a(n)/n!)^(1/n)/n = 1/(exp(1)*(log(2))^2) = 0.7656928576... . - Vaclav Kotesovec, Jun 21 2013
a(n) ~ 2*sqrt(Pi) * n^(2*n-3/2) / (sqrt(1-log(2)) * exp(2*n) * (log(2))^(2*n-1)). - Vaclav Kotesovec, May 13 2014
a(n+1) = Sum_{k = 0..n} A163626(n,k)^2. - Philippe Deléham, May 30 2015
a(n) = A306209(2n-2,n-1). - Alois P. Heinz, Feb 01 2019
a(n) = A266695(2n-2). - Alois P. Heinz, Apr 17 2024

Entry revised by N. J. A. Sloane, Jul 05 2012

A130152 Triangle read by rows: T(n,k) = number of permutations p of [n] such that max(|p(i)-i|)=k (n>=1, 0<=k<=n-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 4, 9, 10, 1, 7, 23, 47, 42, 1, 12, 60, 157, 274, 216, 1, 20, 151, 503, 1227, 1818, 1320, 1, 33, 366, 1669, 4833, 10402, 13656, 9360, 1, 54, 877, 5472, 18827, 50879, 96090, 115080, 75600, 1, 88, 2088, 17531, 75693, 234061, 569602, 966456, 1077840, 685440, 1, 143, 4937, 55135, 304900, 1076807, 3111243, 6791994, 10553640, 11123280, 6894720

Offset: 1

Emeric Deutsch, May 27 2007

Row sums are the factorials. T(n,n) = (n-2)!*(2n-3) = A007680(n-2) (for n>=2). T(n,1) = Fibonacci(n+1)-1 = A000071(n+1). Sum_{k=0..n-1} k*T(n,k) = A130153(n). For the statistic max(p(i)-i) see A056151.

			T(4,1) = 4 because we have 1243, 1324, 2134 and 2143.
Triangle starts:
  1;
  1,  1;
  1,  2,  3;
  1,  4,  9,  10;
  1,  7, 23,  47,  42;
  1, 12, 60, 157, 274, 216;
  ...

Alois P. Heinz, Rows n = 1..23, 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.

Columns k=0-10 give: A000012, A000071(n+1), A323798, A323799, A323800, A323801, A323802, A323803, A323804, A323805, A323806.
Row sums give A000142.
T(n,floor(n/2)) gives A323807.
Cf. A007680, A130153, A056151, A299789, A306209.

with(combinat): for n from 1 to 7 do P:=permute(n): for i from 0 to n-1 do ct[i]:=0 od: for j from 1 to n! do if max(seq(abs(P[j][i]-i),i=1..n))=0 then ct[0]:=ct[0]+1 elif max(seq(abs(P[j][i]-i),i=1..n))=1 then ct[1]:=ct[1]+1 elif max(seq(abs(P[j][i]-i),i=1..n))=2 then ct[2]:=ct[2]+1 elif max(seq(abs(P[j][i]-i),i=1..n))=3 then ct[3]:=ct[3]+1 elif max(seq(abs(P[j][i]-i),i=1..n))=4 then ct[4]:=ct[4]+1 elif max(seq(abs(P[j][i]-i),i=1..n))=5 then ct[5]:=ct[5]+1 elif max(seq(abs(P[j][i]-i),i=1..n))=6 then ct[6]:=ct[6]+1 else fi od: a[n]:=seq(ct[i],i=0..n-1): od: for n from 1 to 7 do a[n] od; # a cumbersome program to obtain, by straightforward counting, the first 7 rows of the triangle
n := 8: st := proc (p) max(seq(abs(p[j]-j), j = 1 .. nops(p))) end proc: with(combinat): P := permute(n): f := sort(add(t^st(P[i]), i = 1 .. factorial(n))); # program gives the row generating polynomial for the specified n - Emeric Deutsch, Aug 13 2009
# second Maple program:
b:= proc(s) option remember; (n-> `if`(n=0, 1, add((p-> add(
      coeff(p, x, i)*x^max(i, abs(n-j)), i=0..degree(p)))(
        b(s minus {j})), j=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b({$1..n})):
seq(T(n), n=1..10);  # Alois P. Heinz, Jan 21 2019
# third Maple program:
A:= proc(n, k) option remember; LinearAlgebra[Permanent](
      Matrix(n, (i, j)-> `if`(abs(i-j)<=k, 1, 0)))
    end:
T:= (n, k)-> A(n, k)-A(n, k-1):
seq(seq(T(n, k), k=0..n-1), n=1..10);  # Alois P. Heinz, Jan 22 2019

(* from second Maple program: *)
b[s_List] := b[s] = Function[n, If[n == 0, 1, Sum[Function[p, Sum[ Coefficient[p, x, i]*x^Max[i, Abs[n - j]], {i, 0, Exponent[p, x]}]][b[s ~Complement~ {j}]], {j, s}]]][Length[s]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n-1}]][b[Range[n]] ];
Table[T[n], {n, 1, 11}] // Flatten
(* from third Maple program: *)
A[n_, k_] := A[n, k] = Permanent[Table[If[Abs[i-j] <= k, 1, 0], {i, 1, n}, {j, 1, n}]];
T[n_, k_] := A[n, k] - A[n, k - 1];
Table[Table[T[n, k], {k, 0, n - 1}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

T(n,k) = A306209(n,k) - A306209(n,k-1) for k > 0, T(n,0) = 1. - Alois P. Heinz, Jan 29 2019

More terms from R. J. Mathar, Oct 15 2007

A154654 Number of permutations of length n within distance 5.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 3720, 17304, 76110, 329462, 1441923, 6487445, 29555588, 135025756, 615260976, 2791161792, 12618600768, 57008446080, 257708989200, 1166042944564, 5279435858788, 23908888017477, 108262665958797, 490132089640318, 2218641353956314

Offset: 0

Torleiv Kløve, Jan 13 2009

a(n) equals the permanent of the n X n matrix with 1's along the eleven central diagonals and 0's everywhere else. - John M. Campbell, Jul 10 2011

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n=1..400 from R. H. Hardin)
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.

Cf. A000045, A002524, A002526, A072856.

Column k=5 of A306209.

G.f. is a rational function f(x)/g(x) where f has degree 132 and g has degree 142.

a(0)=1 prepended by Alois P. Heinz, Jan 28 2019

A154655 Number of permutations of length n within distance 6.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 30960, 172200, 899064, 4553166, 22934774, 116914351, 610093513, 3222826972, 17101449940, 90706002192, 479654768640, 2527274267136, 13280313508416, 69734129749632, 366283822765632, 1925290900630896, 10126754515065868

Offset: 0

Torleiv Kløve, Jan 13 2009

a(n) equals the permanent of the n X n matrix with 1's along the central thirteen diagonals, and 0's everywhere else. - John M. Campbell, Jul 10 2011

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n=1..400 from R. H. Hardin)
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.

Cf. A000045, A002524, A002526, A072856, A154654.

Column k=6 of A306209.

G.f. is a rational function f(x)/g(x) where f has degree 482 and g has degree 494.

a(0)=1 prepended by Alois P. Heinz, Jan 28 2019

A154656 Number of permutations of length n within distance 7.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 287280, 1865520, 11345160, 66349464, 381523758, 2193664790, 12764590275, 75796724309, 455383613924, 2750869551868, 16635586999056, 100439873614656, 604666567043712, 3629299734118656, 21736009354060800, 130082373922081536

Offset: 0

Torleiv Kløve, Jan 13 2009

a(n) equals the permanent of the n X n matrix with 1's along the central fifteen diagonals, and 0's everywhere else. - John M. Campbell, Jul 10 2011

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n=1..400 from R. H. Hardin)
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.

Cf. A000045, A002524, A002526, A072856, A154654.

Column k=7 of A306209.

a(0)=1 prepended by Alois P. Heinz, Jan 28 2019

cp's OEIS Frontend

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

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A072856 Number of permutations satisfying i-4<=p(i)<=i+4, i=1..n (permutations of length n within distance 4).

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A001564 2nd differences of factorial numbers.

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A002526 Number of permutations of length n within distance 3 of a fixed permutation.

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A092552 Let X_{m,n}(q) be the chromatic polynomial of the complete bipartite graph K_{m,n}. Then a(n) is the negative of the coefficient of the linear term of X_{n,n}(q).

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A048163 a(n) = Sum_{k=1..n} ((k-1)!)^2*Stirling2(n,k)^2.

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