A049774 -id:A049774 - cp's OEIS frontend

A177553 Number of permutations of 1..n avoiding adjacent step pattern up, up, up, up, up, up.

1, 1, 2, 6, 24, 120, 720, 5039, 40305, 362682, 3626190, 39881160, 478490760, 6219298800, 87055051511, 1305598835941, 20885951018102, 354999461960226, 6388879812001704, 121367620532150280, 2426930566055020080, 50956684690331669759, 1120852238721212726609

Offset: 0

R. H. Hardin, May 10 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450
Mingjia Yang, Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018.
Mingjia Yang, An experimental walk in patterns, partitions, and words, Ph. D. Dissertation, Rutgers University (2020).

Column k=63 of A242784.

Cf. A080635, A049774, A117158, A177533, A177523.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<5, add(b(u+j-1, o-j, t+1), j=1..o), 0)+
      add(b(u-j, o+j-1, 0), j=1..u))
    end:
a:= n-> b(n, 0, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 07 2013

nn=20;r=6;a=Apply[Plus,Table[Normal[Series[y x^(r+1)/(1-Sum[y x^i,{i,1,r}]),{x,0,nn}]][[n]]/(n+r)!,{n,1,nn-r}]]/.y->-1;Range[0,nn]! CoefficientList[Series[1/(1-x-a),{x,0,nn}],x] (* Geoffrey Critzer, Feb 25 2014 *)
Table[n!*SeriesCoefficient[1/(Sum[x^(7*k)/(7*k)!-x^(7*k+1)/(7*k+1)!,{k,0,n}]),{x,0,n}],{n,1,20}] (* Vaclav Kotesovec, Aug 29 2014 *)

a(n)/n! ~ c * (1/r)^n, where r = 1.0001738181531504504518260962714687775785823593018886... is the root of the equation Sum_{n>=0} (r^(7*n)/(7*n)! - r^(7*n+1)/(7*n+1)!) = 0, c = 1.0010191104259450282450770594076722424772755532278.... - Vaclav Kotesovec, Aug 29 2014
E.g.f.: -(7/(2*((-cos(x*cos(3*Pi/14)))*cosh(x*sin(3*Pi/14)) + cos(x*cos(3*Pi/14))*cosh(x*sin(3*Pi/14))* sin(3*Pi/14) - cosh(x*sin(Pi/14))* (cos(x*cos(Pi/14))*(1 + sin(Pi/14)) - cos(Pi/14)*sin(x*cos(Pi/14))) + cos(3*Pi/14)*cosh(x*sin(3*Pi/14))* sin(x*cos(3*Pi/14)) - cosh(x*cos(Pi/7))* ((1 + cos(Pi/7))*cos(x*sin(Pi/7)) - sin(Pi/7)*sin(x*sin(Pi/7))) + cos(x*sin(Pi/7))* sinh(x*cos(Pi/7)) + cos(Pi/7)*cos(x*sin(Pi/7))* sinh(x*cos(Pi/7)) - sin(Pi/7)*sin(x*sin(Pi/7))* sinh(x*cos(Pi/7)) + cos(x*cos(Pi/14))* sinh(x*sin(Pi/14)) + cos(x*cos(Pi/14))*sin(Pi/14)* sinh(x*sin(Pi/14)) - cos(Pi/14)*sin(x*cos(Pi/14))* sinh(x*sin(Pi/14)) - cos(x*cos(3*Pi/14))* sinh(x*sin(3*Pi/14)) + cos(x*cos(3*Pi/14))* sin(3*Pi/14)*sinh(x*sin(3*Pi/14)) + cos(3*Pi/14)*sin(x*cos(3*Pi/14))* sinh(x*sin(3*Pi/14))))). - Vaclav Kotesovec, Jan 31 2015
In closed form, c = 7 / (r * (2*cos(r*sin(Pi/7))*cosh(r*cos(Pi/7)) + cos(Pi/7 - r*sin(Pi/7)) * cosh(r*cos(Pi/7)) + cos(Pi/7 - r*sin(Pi/7)) * cosh(r*cos(Pi/7)) + 2*cos(r*cos(Pi/14)) * cosh(r*sin(Pi/14)) + 2*cos(r*cos(3*Pi/14)) * cosh(r*sin(3*Pi/14)) + 2*cosh(r*sin(Pi/14)) * sin(Pi/14 + r*cos(Pi/14)) - 2*cosh(r*sin(3*Pi/14)) * sin(3*Pi/14 - r*cos(3*Pi/14)) - 2*cos(r*sin(Pi/7)) * sinh(r*cos(Pi/7)) - cos(Pi/7 - r*sin(Pi/7)) * sinh(r*cos(Pi/7)) - cos(Pi/7 - r*sin(Pi/7)) * sinh(r*cos(Pi/7)) - 2*cos(r*cos(Pi/14)) * sinh(r*sin(Pi/14)) - 2*sin(Pi/14 + r*cos(Pi/14))*sinh(r*sin(Pi/14)) + 2*cos(r*cos(3*Pi/14)) * sinh(r*sin(3*Pi/14)) - 2*sin((3*Pi)/14 - r*cos(3*Pi/14)) * sinh(r*sin(3*Pi/14)))). - Vaclav Kotesovec, Feb 01 2015

a(18)-a(22) from Alois P. Heinz, Oct 07 2013

a(0)=1 prepended by Alois P. Heinz, Aug 08 2018

A000303 Number of permutations of [n] in which the longest increasing run has length 2.

Original entry on oeis.org

0, 1, 4, 16, 69, 348, 2016, 13357, 99376, 822040, 7477161, 74207208, 797771520, 9236662345, 114579019468, 1516103040832, 21314681315997, 317288088082404, 4985505271920096, 82459612672301845, 1432064398910663704, 26054771465540507272

Offset: 1

N. J. A. Sloane

nonn

			a(3)=4 because we have (13)2, 2(13), (23)1, 3(12), where the parentheses surround increasing runs of length 2.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261, Table 7.4.1.
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).

Alois P. Heinz, Table of n, a(n) for n = 1..464 (first 100 terms from Max Alekseyev)
Max A. Alekseyev, On the number of permutations with bounded runs length, arXiv preprint arXiv:1205.4581 [math.CO], 2012-2013. - From _N. J. A. Sloane_, Oct 23 2012

Column 2 of A008304. Other columns: A000402, A000434, A000456, A000467, A230055.
Cf. A001250, A001251, A001252, A001253, A010026, A211318.
Equals 1 less than A049774. - Greg Dresden, Feb 22 2020

b[u_, o_, t_, k_] := b[u, o, t, k] = If[t == k, (u + o)!, If[Max[t, u] + o < k, 0, Sum[b[u + j - 1, o - j, t + 1, k], {j, 1, o}] + Sum[b[u - j, o + j - 1, 1, k], {j, 1, u}]]];
T[n_, k_] := b[0, n, 0, k] - b[0, n, 0, k + 1];
a[n_] := T[n, 2];
Array[a, 30] (* Jean-François Alcover, Jul 19 2018, after Alois P. Heinz *)

Better description from Emeric Deutsch, May 08 2004

Edited and extended by Max Alekseyev, May 20 2012

A230231 Number of permutations of [n] avoiding adjacent step pattern {up}^8.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362879, 3628781, 39916492, 478996716, 6226941864, 87176969880, 1307651304960, 20922368987520, 355679390626560, 6402213152423659, 121641748198554547, 2432828930036156696, 51089280818439941448, 1123961390341566969192

Offset: 0

Alois P. Heinz, Oct 12 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450
Mingjia Yang, Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018.

Cf. A049774, A117158, A177523, A177533, A177553, A230051, A230232, A230233.

Column k=255 of A242784.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<7, add(b(u+j-1, o-j, t+1), j=1..o), 0)+
      add(b(u-j, o+j-1, 0), j=1..u))
    end:
a:= n-> b(n, 0, 0):
seq(a(n), n=0..30);

nn=20;r=8;a=Apply[Plus,Table[Normal[Series[y x^(r+1)/(1-Sum[y x^i,{i,1,r}]),{x,0,nn}]][[n]]/(n+r)!,{n,1,nn-r}]]/.y->-1;Range[0,nn]! CoefficientList[Series[1/(1-x-a),{x,0,nn}],x] (* Geoffrey Critzer, Feb 25 2014 *)
CoefficientList[Series[1/(HypergeometricPFQ[{}, {1/9, 2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 8/9}, x^9/387420489] - x*HypergeometricPFQ[{}, {2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 8/9, 10/9}, x^9/387420489]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 01 2015 *)

E.g.f.: 1 / Sum_{n>=0} (9*n+1-x)*x^(9*n)/(9*n+1)!.
a(n)/n! ~ 1.0000195665100891649606434859189953881417919885320660432331680939719... * (1/r)^n, where r = 1.0000024802134092668222044475851121972165291678378389183730077680957571... is the root of the equation Sum_{n>=0} (r^(9*n)/(9*n)! - r^(9*n+1)/(9*n+1)!) = 0. - Vaclav Kotesovec, Jan 17 2015
E.g.f.: 1/(1/3 * cos((sqrt(3)*x)/2) * cosh(x/2) + 2/9 * cos(x * sin(Pi/9)) * cosh(x * cos(Pi/9)) + 2/9 * cos(Pi/9) * cos(x * sin(Pi/9)) * cosh(x * cos(Pi/9)) + 2/9 * cos(x * cos(Pi/18)) * cosh(x * sin(Pi/18)) + 2/9 * cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9)))* cosh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))) - 1/9 * cos(Pi/9) * cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9))) * cosh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))) - 2/9 * cos(x * cos(Pi/18))* cosh(x * sin(Pi/18)) * sin(Pi/18) - (cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9)))* cosh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9)))* sin(Pi/9))/(3 * sqrt(3)) - (cosh(x/2) * sin((sqrt(3)*x)/2))/(3 * sqrt(3)) - 2/9 * cos(Pi/18) * cosh(x * sin(Pi/18)) * sin(x * cos(Pi/18)) - (cos(Pi/9) * cosh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9)))* sin(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9))))/ (3 * sqrt(3)) + 1/9 * cosh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9)))* sin(Pi/9) * sin(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9))) - 2/9 * cosh(x * cos(Pi/9)) * sin(Pi/9)* sin(x * sin(Pi/9)) - 1/3 * cos((sqrt(3)*x)/2)* sinh(x/2) + (sin((sqrt(3)*x)/2) * sinh(x/2))/ (3 * sqrt(3)) - 2/9 * cos(x * sin(Pi/9))* sinh(x * cos(Pi/9)) - 2/9 * cos(Pi/9) * cos(x * sin(Pi/9))* sinh(x * cos(Pi/9)) + 2/9 * sin(Pi/9) * sin(x * sin(Pi/9))* sinh(x * cos(Pi/9)) + 2/9 * cos(x * cos(Pi/18))* sinh(x * sin(Pi/18)) - 2/9 * cos(x * cos(Pi/18))* sin(Pi/18) * sinh(x * sin(Pi/18)) - 2/9 * cos(Pi/18)* sin(x * cos(Pi/18)) * sinh(x * sin(Pi/18)) + 2/9 * cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9)))* sinh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))) - 1/9 * cos(Pi/9) * cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9))) * sinh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))) - (cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9))) * sin(Pi/9)* sinh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))))/ (3 * sqrt(3)) - (cos(Pi/9) * sin(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9)))* sinh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))))/ (3 * sqrt(3)) + 1/9 * sin(Pi/9)* sin(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9)))* sinh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9)))). - Vaclav Kotesovec, Feb 01 2015

A230232 Number of permutations of [n] avoiding adjacent step pattern {up}^9.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628799, 39916779, 479001228, 6227014404, 87178179816, 1307672369640, 20922752672640, 355686706327680, 6402359109968640, 121644792614741760, 2432895242801771955, 51090787299486057355, 1123997039003038423610

Offset: 0

Alois P. Heinz, Oct 12 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A049774, A117158, A177523, A177533, A177553, A230051, A230231, A230233.

Column k=511 of A242784.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<8, add(b(u+j-1, o-j, t+1), j=1..o), 0)+
      add(b(u-j, o+j-1, 0), j=1..u))
    end:
a:= n-> b(n, 0, 0):
seq(a(n), n=0..30);

nn=20;r=9;a=Apply[Plus,Table[Normal[Series[y x^(r+1)/(1-Sum[y x^i,{i,1,r}]),{x,0,nn}]][[n]]/(n+r)!,{n,1,nn-r}]]/.y->-1;Range[0,nn]! CoefficientList[Series[1/(1-x-a),{x,0,nn}],x] (* Geoffrey Critzer, Feb 25 2014 *)
FullSimplify[CoefficientList[Series[10/(2/E^x - Sqrt[2*(5 - Sqrt[5])]* Cosh[(1/4)*(1 + Sqrt[5])*x]* Sin[Sqrt[(1/8)*(5 - Sqrt[5])]*x] - Sqrt[2*(5 + Sqrt[5])]*Cosh[(1/4)*(Sqrt[5] - 1)* x]*Sin[Sqrt[(1/8)*(5 + Sqrt[5])]*x] + Cos[Sqrt[(1/8)*(5 + Sqrt[5])]*x]* (4*Cosh[(1/4)*(Sqrt[5] - 1)*x] - (Sqrt[5] - 1)*Sinh[(1/4)*(Sqrt[5] - 1)*x]) - Cos[Sqrt[(1/8)*(5 - Sqrt[5])]*x]* ((1 + Sqrt[5])*Sinh[(1/4)*(1 + Sqrt[5])*x] - 4*Cosh[(1/4)*(1 + Sqrt[5])*x])), {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 31 2015 *)

E.g.f.: 1 / Sum_{n>=0} (10*n+1-x)*x^(10*n)/(10*n+1)!.
a(n)/n! ~ c * (1/r)^n, where r = 1.0000002505217051890946793081039639693008257169189079028339632923816... is the root of the equation Sum_{n>=0} (r^(10*n)/(10*n)! - r^(10*n+1)/(10*n+1)!) = 0, c = 1.000002229648140602899529055054469878816530201510267349345270187155... . - Vaclav Kotesovec, Jan 17 2015
E.g.f.: 10 / (2/exp(x) - sqrt(2*(5 - sqrt(5))) * cosh((1/4)*(1 + sqrt(5))*x) * sin(sqrt((1/8)*(5 - sqrt(5)))*x) - sqrt(2*(5 + sqrt(5))) * cosh((1/4)*(sqrt(5) - 1)*x) * sin(sqrt((1/8)*(5 + sqrt(5)))*x) + cos(sqrt((1/8)*(5 + sqrt(5)))*x) * (4*cosh((1/4)*(sqrt(5) - 1)*x) - (sqrt(5) - 1)*sinh((1/4)*(sqrt(5) - 1)*x)) - cos(sqrt((1/8)*(5 - sqrt(5)))*x) * ((1 + sqrt(5))*sinh((1/4)*(1 + sqrt(5))*x) - 4*cosh((1/4)*(1 + sqrt(5))*x))). - Vaclav Kotesovec, Jan 31 2015
In closed form, c = 5 / (r * (sqrt(10 - 2*sqrt(5)) * cosh((sqrt(5)+1)*r/4) * sin(sqrt((5 - sqrt(5))/2)*r/2) + sqrt(2*(5 + sqrt(5))) * cosh((sqrt(5)-1)*r/4) * sin(sqrt((5 + sqrt(5))/2)*r/2))). - Vaclav Kotesovec, Feb 01 2015

A263643 T(n,k)=Number of length n arrays of permutations of 0..n-1 with each element moved by -k to k places and with no two consecutive increases.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 5, 2, 1, 2, 5, 9, 2, 1, 2, 5, 17, 11, 2, 1, 2, 5, 17, 41, 19, 2, 1, 2, 5, 17, 70, 75, 27, 2, 1, 2, 5, 17, 70, 226, 156, 44, 2, 1, 2, 5, 17, 70, 349, 538, 340, 65, 2, 1, 2, 5, 17, 70, 349, 1389, 1417, 738, 104, 2, 1, 2, 5, 17, 70, 349, 2017, 4255, 3734, 1567, 155, 2

Offset: 1

R. H. Hardin, Oct 22 2015

Table starts
.1...1....1.....1.....1......1......1......1......1......1......1......1......1
.2...2....2.....2.....2......2......2......2......2......2......2......2......2
.2...5....5.....5.....5......5......5......5......5......5......5......5......5
.2...9...17....17....17.....17.....17.....17.....17.....17.....17.....17.....17
.2..11...41....70....70.....70.....70.....70.....70.....70.....70.....70.....70
.2..19...75...226...349....349....349....349....349....349....349....349....349
.2..27..156...538..1389...2017...2017...2017...2017...2017...2017...2017...2017
.2..44..340..1417..4255...9673..13358..13358..13358..13358..13358..13358..13358
.2..65..738..3734.13529..36321..74678..99377..99377..99377..99377..99377..99377
.2.104.1567.10564.42700.138420.335720.636645.822041.822041.822041.822041.822041

			Some solutions for n=6 k=4
..2....2....3....0....2....3....3....4....1....4....1....4....4....0....4....1
..0....1....0....5....5....1....4....0....4....5....5....2....3....5....1....0
..4....4....2....1....3....5....2....2....0....2....4....5....5....2....0....4
..1....0....1....3....0....0....5....1....3....0....0....3....0....3....5....3
..5....5....5....2....4....4....0....5....2....3....3....0....2....1....2....5
..3....3....4....4....1....2....1....3....5....1....2....1....1....4....3....2

R. H. Hardin, Table of n, a(n) for n = 1..484

Diagonal is A049774.

Empirical for column k:
k=2: a(n) = 2*a(n-2) +a(n-3) -a(n-5) for n>9
k=3: [order 14] for n>19
k=4: [order 46] for n>53

A230233 Number of permutations of [n] avoiding adjacent step pattern {up}^10.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916799, 479001577, 6227020358, 87178283010, 1307674215120, 20922786961440, 355687370176320, 6402372516146880, 121645075013280000, 2432901444395385600, 51090929159028595200, 1124000415686590747031

Offset: 0

Alois P. Heinz, Oct 12 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A049774, A117158, A177523, A177533, A177553, A230051, A230231, A230232.

Column k=1023 of A242784.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<9, add(b(u+j-1, o-j, t+1), j=1..o), 0)+
      add(b(u-j, o+j-1, 0), j=1..u))
    end:
a:= n-> b(n, 0, 0):
seq(a(n), n=0..30);

nn=20;r=10;a=Apply[Plus,Table[Normal[Series[y x^(r+1)/(1-Sum[y x^i,{i,1,r}]),{x,0,nn}]][[n]]/(n+r)!,{n,1,nn-r}]]/.y->-1;Range[0,nn]! CoefficientList[Series[1/(1-x-a),{x,0,nn}],x] (* Geoffrey Critzer, Feb 25 2014 *)
CoefficientList[Series[1/(HypergeometricPFQ[{}, {1/11, 2/11, 3/11, 4/11, 5/11, 6/11, 7/11, 8/11, 9/11, 10/11}, x^11/285311670611] - x*HypergeometricPFQ[{}, {2/11, 3/11, 4/11, 5/11, 6/11, 7/11, 8/11, 9/11, 10/11, 12/11}, x^11/285311670611]), {x, 0, 25}], x] * Range[0, 25]! (* Vaclav Kotesovec, Jan 17 2015 *)

E.g.f.: 1 / Sum_{n>=0} (11*n+1-x)*x^(11*n)/(11*n+1)!.
a(n)/n! ~ 1.000000227556759905306252970186381144189779110025896440589711080508... * (1/r)^n, where r = 1.000000022964438439732421879840792836238519233492197325926442472620564... is the root of the equation Sum_{n>=0} (r^(11*n)/(11*n)! - r^(11*n+1)/(11*n+1)!) = 0. - Vaclav Kotesovec, Jan 17 2015
E.g.f.: -11/(2*((-cos(x*cos(Pi/22)))* cosh(x*sin(Pi/22)) - cos(x*cos(3*Pi/22))* cosh(x*sin(3*Pi/22)) - cos(x*cos(5*Pi/22))* cosh(x*sin(5*Pi/22)) - cos(x*cos(Pi/22))* cosh(x*sin(Pi/22))*sin(Pi/22) + cos(x*cos(3*Pi/22))* cosh(x*sin(3*Pi/22))* sin(3*Pi/22) - cos(x*cos(5*Pi/22))* cosh(x*sin(5*Pi/22))* sin(5*Pi/22) + cos(Pi/22)* cosh(x*sin(Pi/22))* sin(x*cos(Pi/22)) + cos(3*Pi/22)*cosh( x*sin(3*Pi/22))* sin(x*cos(3*Pi/22)) + cos(5*Pi/22)*cosh( x*sin(5*Pi/22))* sin(x*cos(5*Pi/22)) - cosh(x*cos(Pi/11))* ((1 + cos(Pi/11))* cos(x*sin(Pi/11)) - sin(Pi/11)*sin(x*sin(Pi/11))) + cosh(x*cos(2*Pi/11))* ((-1 + cos(2*Pi/11))* cos(x*sin(2*Pi/11)) + sin(2*Pi/11)* sin(x*sin(2*Pi/11))) + cos(x*sin(Pi/11))* sinh(x*cos(Pi/11)) + cos(Pi/11)*cos(x*sin(Pi/11))* sinh(x*cos(Pi/11)) - sin(Pi/11)*sin(x*sin(Pi/11))* sinh(x*cos(Pi/11)) - cos(x*sin(2*Pi/11))* sinh(x*cos(2*Pi/11)) + cos(2*Pi/11)* cos(x*sin(2*Pi/11))* sinh(x*cos(2*Pi/11)) + sin(2*Pi/11)* sin(x*sin(2*Pi/11))* sinh(x*cos(2*Pi/11)) + cos(x*cos(Pi/22))* sinh(x*sin(Pi/22)) + cos(x*cos(Pi/22))*sin(Pi/22)* sinh(x*sin(Pi/22)) - cos(Pi/22)*sin(x*cos(Pi/22))* sinh(x*sin(Pi/22)) - cos(x*cos(3*Pi/22))* sinh(x*sin(3*Pi/22)) + cos(x*cos(3*Pi/22))* sin(3*Pi/22)* sinh(x*sin(3*Pi/22)) + cos(3*Pi/22)* sin(x*cos(3*Pi/22))* sinh(x*sin(3*Pi/22)) + cos(x*cos(5*Pi/22))* sinh(x*sin(5*Pi/22)) + cos(x*cos(5*Pi/22))* sin(5*Pi/22)* sinh(x*sin(5*Pi/22)) - cos(5*Pi/22)* sin(x*cos(5*Pi/22))* sinh(x*sin(5*Pi/22)))). - Vaclav Kotesovec, Jan 31 2015

A263683 T(n,k)=Number of length n arrays of permutations of 0..n-1 with each element moved by -k to k places and with no two consecutive decreases.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 5, 5, 1, 2, 5, 12, 8, 1, 2, 5, 17, 26, 13, 1, 2, 5, 17, 53, 58, 21, 1, 2, 5, 17, 70, 155, 131, 34, 1, 2, 5, 17, 70, 277, 429, 295, 55, 1, 2, 5, 17, 70, 349, 1009, 1210, 662, 89, 1, 2, 5, 17, 70, 349, 1658, 3487, 3457, 1487, 144, 1, 2, 5, 17, 70, 349, 2017

Offset: 1

R. H. Hardin, Oct 23 2015

Table starts
..1....1....1.....1......1......1......1......1......1......1......1......1
..2....2....2.....2......2......2......2......2......2......2......2......2
..3....5....5.....5......5......5......5......5......5......5......5......5
..5...12...17....17.....17.....17.....17.....17.....17.....17.....17.....17
..8...26...53....70.....70.....70.....70.....70.....70.....70.....70.....70
.13...58..155...277....349....349....349....349....349....349....349....349
.21..131..429..1009...1658...2017...2017...2017...2017...2017...2017...2017
.34..295.1210..3487...7356..11253..13358..13358..13358..13358..13358..13358
.55..662.3457.11708..30374..58743..85403..99377..99377..99377..99377..99377
.89.1487.9948.39905.121676.286327.514631.717372.822041.822041.822041.822041

			Some solutions for n=7 k=4
..2....2....0....0....0....4....1....1....1....3....0....2....2....1....0....2
..0....3....4....2....1....0....5....3....2....4....5....4....1....4....4....0
..3....0....3....1....4....1....2....6....0....5....1....0....5....6....5....1
..4....4....6....4....3....3....3....0....4....6....2....6....0....0....1....6
..5....1....1....5....5....6....0....2....6....0....3....1....4....2....6....3
..1....6....2....6....6....2....4....5....3....1....4....5....6....3....2....5
..6....5....5....3....2....5....6....4....5....2....6....3....3....5....3....4

R. H. Hardin, Table of n, a(n) for n = 1..484

Column 1 is A000045(n+1).
Column 2 is A116716.
Diagonal is A049774.

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = 2*a(n-1) +a(n-3) +a(n-4) -a(n-5)
k=3: [order 20]
k=4: [order 70]

A350355 Numbers k such that the k-th composition in standard order is up/down.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 13, 16, 20, 24, 25, 32, 40, 41, 48, 49, 50, 54, 64, 72, 80, 81, 82, 96, 97, 98, 102, 108, 109, 128, 144, 145, 160, 161, 162, 166, 192, 193, 194, 196, 198, 204, 205, 216, 217, 256, 272, 288, 289, 290, 320, 321, 322, 324, 326, 332, 333, 384

Offset: 1

Gus Wiseman, Jan 15 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

A composition is up/down if it is alternately strictly increasing and strictly decreasing, starting with an increase. For example, the partition (3,2,2,2,1) has no up/down permutations, even though it does have the anti-run permutation (2,3,2,1,2).

			The terms together with the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   6: (1,2)
   8: (4)
  12: (1,3)
  13: (1,2,1)
  16: (5)
  20: (2,3)
  24: (1,4)
  25: (1,3,1)
  32: (6)
  40: (2,4)
  41: (2,3,1)
  48: (1,5)
  49: (1,4,1)
  50: (1,3,2)
  54: (1,2,1,2)

The case of permutations is counted by A000111.
These compositions are counted by A025048, down/up A025049.
The strict case is counted by A129838, undirected A349054.
The weak version is counted by A129852, down/up A129853.
The version for anti-runs is A333489, a superset, complement A348612.
This is the up/down case of A345167, counted by A025047.
Counting patterns of this type gives A350354.
The down/up version is A350356.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A011782 counts compositions, unordered A000041.
A345192 counts non-alternating compositions, ranked by A345168.
A349052 counts weakly alternating compositions, complement A349053.
A349057 ranks non-weakly alternating compositions.
Statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of maximal anti-runs is A333381.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Patterns are A333217.
Cf. A008965, A049774, A095684, A106356, A238279, A344604, A344614, A344615, A345169, A345170, A345172, A349051, A349799.

updoQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]>y[[m+1]],y[[m]]

A345167 = A350355 \/ A350356.

A350356 Numbers k such that the k-th composition in standard order is down/up.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 16, 17, 18, 22, 32, 33, 34, 38, 44, 45, 64, 65, 66, 68, 70, 76, 77, 88, 89, 128, 129, 130, 132, 134, 140, 141, 148, 152, 153, 176, 177, 178, 182, 256, 257, 258, 260, 262, 264, 268, 269, 276, 280, 281, 296, 297, 304, 305, 306, 310, 352, 353

Offset: 1

Gus Wiseman, Jan 15 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

A composition is down/up if it is alternately strictly increasing and strictly decreasing, starting with a decrease. For example, the partition (3,2,2,2,1) has no down/up permutations, even though it does have the anti-run permutation (2,1,2,3,2).

			The terms together with the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   5: (2,1)
   8: (4)
   9: (3,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  22: (2,1,2)
  32: (6)
  33: (5,1)
  34: (4,2)
  38: (3,1,2)
  44: (2,1,3)
  45: (2,1,2,1)

Gus Wiseman, Statistics, classes, and transformations of standard compositions
Wikipedia, Alternating permutation

The case of permutations is counted by A000111.
These compositions are counted by A025049, up/down A025048.
The strict case is counted by A129838, undirected A349054.
The weak version is counted by A129853, up/down A129852.
The version for anti-runs is A333489, a superset, complement A348612.
This is the down/up case of A345167, counted by A025047.
Counting patterns of this type gives A350354.
The up/down version is A350355.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A011782 counts compositions, unordered A000041.
A345192 counts non-alternating compositions, ranked by A345168.
A349052 counts weakly alternating compositions, complement A349053.
A349057 ranks non-weakly alternating compositions.
Statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of maximal anti-runs is A333381.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Patterns are A333217.
Cf. A008965, A049774, A095684, A106356, A238279, A344604, A344614, A344615, A345169, A345170, A345172, A349051, A349799.

doupQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]y[[m+1]]],{m,1,Length[y]-1}];
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],doupQ[stc[#]]&]

A345167 = A350355 \/ A350356.

A071076 Number of permutations that avoid the generalized pattern 123-4.

Original entry on oeis.org

1, 1, 2, 6, 23, 108, 598, 3815, 27532, 221708, 1970251, 19150132, 202064380, 2300071071, 28092017668, 366425723926, 5083645400819, 74745472084176, 1160974832572274, 18995175706664735, 326531476287842760, 5883736110875887560, 110893188848753125475

Offset: 0

Sergey Kitaev, May 26 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450
A. M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011.
Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv preprint arXiv:1108.2642 [math.CO], 2011.
Sergey Kitaev, Partially Ordered Generalized Patterns, preprint.
Sergey Kitaev, Partially Ordered Generalized Patterns, Discrete Math. 298 (2005), no. 1-3, 212-229.
Yan Wang, Qi Fang, Shishuo Fu, Sergey Kitaev, and Haijun Li, Consecutive and quasi-consecutive patterns: des-Wilf classifications and generating functions, arXiv:2502.10128 [math.CO], 2025. See p. 9.

Cf. A049774, A071088, A071075, A071077.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(
      `if`(t=1 and o>j, 0, b(u+j-1, o-j, t+1)), j=1..o)+
       add(b(u-j, o+j-1, 0), j=1..u))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 14 2015

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Sum[If[t == 1 && o > j, 0, b[u + j - 1, o - j, t + 1]], {j, 1, o}] + Sum[b[u - j, o + j - 1, 0], {j, 1, u}]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 01 2016, after Alois P. Heinz *)

E.g.f.: exp(int(A(y), y=0..x)), where A(y) = (sqrt(3)/2)*exp(y/2)/cos((sqrt(3)/2)*y + Pi/6).

Let b(n) = A049774(n) = number of permutations of [n] that avoid the consecutive pattern 123. Then a(n) = Sum_{i = 0..n-1} binomial(n-1,i)*b(i)*a(n-1-i) with a(0) = b(0) = 1. [See the recurrence for A_n and B_n in the proof of Theorem 13 in Kitaev's papers.] -

More terms from Vladeta Jovovic, May 28 2002
Link added by Andrew Baxter, May 17 2011
Typos in formula corrected by Vaclav Kotesovec, Aug 23 2014

cp's OEIS Frontend

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A263643 T(n,k)=Number of length n arrays of permutations of 0..n-1 with each element moved by -k to k places and with no two consecutive increases.

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A230233 Number of permutations of [n] avoiding adjacent step pattern {up}^10.

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A263683 T(n,k)=Number of length n arrays of permutations of 0..n-1 with each element moved by -k to k places and with no two consecutive decreases.

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A350355 Numbers k such that the k-th composition in standard order is up/down.

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