A291684 -id:A291684 - cp's OEIS frontend

A303697 Number T(n,k) of permutations p of [n] whose difference between sum of up-jumps and sum of down-jumps equals k; triangle T(n,k), n>=0, min(0,1-n)<=k<=max(0,n-1), read by rows.

1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 4, 5, 4, 5, 4, 1, 1, 11, 19, 19, 20, 19, 19, 11, 1, 1, 26, 82, 100, 101, 100, 101, 100, 82, 26, 1, 1, 57, 334, 580, 619, 619, 620, 619, 619, 580, 334, 57, 1, 1, 120, 1255, 3394, 4339, 4420, 4421, 4420, 4421, 4420, 4339, 3394, 1255, 120, 1

Offset: 0

Alois P. Heinz, Apr 28 2018

An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.

			Triangle T(n,k) begins:
:                               1                             ;
:                               1                             ;
:                          1,   0,   1                        ;
:                     1,   1,   2,   1,   1                   ;
:                1,   4,   5,   4,   5,   4,   1              ;
:           1,  11,  19,  19,  20,  19,  19,  11,   1         ;
:      1,  26,  82, 100, 101, 100, 101, 100,  82,  26,  1     ;
:  1, 57, 334, 580, 619, 619, 620, 619, 619, 580, 334, 57, 1  ;

Alois P. Heinz, Rows n = 0..125, flattened

Row sums give A000142.

Cf. A000295, A001720, A005165, A008292, A081285, A153229, A291680, A291684, A291722, A316292, A316293, A321316.

b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
      add(b(u-j, o+j-1)*x^(-j), j=1..u)+
      add(b(u+j-1, o-j)*x^( j), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(
        `if`(n=0, 1, add(b(j-1, n-j), j=1..n))):
seq(T(n), n=0..12);

b[u_, o_] := b[u, o] = Expand[If[u+o == 0, 1,
     Sum[b[u-j, o+j-1] x^-j, {j, 1, u}] +
     Sum[b[u+j-1, o-j] x^j, {j, 1, o}]]];
T[0] = {1};
T[n_] := x^n Sum[b[j-1, n-j], {j, 1, n}] // CoefficientList[#, x]& // Rest;
T /@ Range[0, 12] // Flatten (* Jean-François Alcover, Feb 20 2021, after Alois P. Heinz *)

T(n,0) = A153229(n) for n > 0.
T(n,1) = A005165(n-1) for n > 0.
T(n+1,n-1) = A000295(n).
T(n,k) = T(n,-k).
Sum_{k=0..n-1} k^2 * T(n,k) = A001720(n+2) for n>1.

A291685 Number of permutations p of [n] such that 0p has a nonincreasing jump sequence.

Original entry on oeis.org

1, 1, 2, 5, 16, 52, 189, 683, 2621, 10061, 40031, 159201, 650880, 2657089, 11062682, 46065143, 194595138, 822215099, 3513875245, 15021070567, 64785349064, 279575206629, 1214958544538, 5283266426743, 23106210465665, 101120747493793, 444614706427665

Offset: 0

Alois P. Heinz, Aug 29 2017

nonn

An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.

			a(3) = 5 = 6 - 1 counts all permutations of {1,2,3} except 132 with jump sequence 1, 2, 1.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Row sums and main diagonal (shifted) of A291684.

Cf. A288910, A288911, A288912.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1, j), j=1..min(t, u))+
      add(b(u+j-1, o-j, j), j=1..min(t, o)))
    end:
a:= n-> b(0, n$2):
seq(a(n), n=0..30);

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1,
     Sum[b[u-j, o+j-1, j], {j, Min[t, u]}]+
     Sum[b[u+j-1, o-j, j], {j, Min[t, o]}]];
a[n_] := b[0, n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)

A303204 Number of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with ceiling(n/2).

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 36, 98, 327, 988, 3392, 10872, 38795, 129520, 469662, 1609176, 5935728, 20786804, 77416352, 274792342, 1035050705, 3719296036, 14094000938, 51119572738, 195075365778, 712918642042, 2734475097609, 10055531355652, 38747262233793

Offset: 0

Alois P. Heinz, Apr 19 2018

nonn

An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Bisections give A291688 (even part), A303203 (odd part).

Cf. A291684.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(sort([u-j, o+j-1])[], j), j=1..min(t, u))+
      add(b(sort([u+j-1, o-j])[], j), j=1..min(t, o)))
    end:
a:= n-> `if`(n=0, 1, (j-> b(0, n, j)-b(0, n, j-1))(ceil(n/2))):
seq(a(n), n=0..30);

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1,
     Sum[b[Sequence @@ Sort[{u-j, o+j-1}], j], {j, Min[t, u]}]+
     Sum[b[Sequence @@ Sort[{u+j-1, o-j}], j], {j, Min[t, o]}]];
a[n_] := If[n == 0, 1,
     Function[j, b[0, n, j] - b[0, n, j-1]][Ceiling[n/2]]];
Table[a[n], {n, 0, 30}]; (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)

a(n) = A291684(n,ceiling(n/2)).

A291688 Number of permutations p of [2n] such that 0p has a nonincreasing jump sequence beginning with n.

Original entry on oeis.org

1, 1, 5, 36, 327, 3392, 38795, 469662, 5935728, 77416352, 1035050705, 14094000938, 195075365778, 2734475097609, 38747262233793, 554199475506095, 7990492729051526, 115995691148658656, 1694340616136589743, 24882428969673439384, 367160435328847044586

Offset: 0

Alois P. Heinz, Aug 29 2017

nonn

An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.

			a(2) = 5: 2134, 2314, 2341, 2413, 2431.

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A291684.

Bisection (even part) of A303204.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1, j), j=1..min(t, u))+
      add(b(u+j-1, o-j, j), j=1..min(t, o)))
    end:
a:= n-> b(0, 2*n, n)-`if`(n=0, 0, b(0, 2*n, n-1)):
seq(a(n), n=0..25);

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
     Sum[b[u - j, o + j - 1, j], {j, Min[t, u]}] +
     Sum[b[u + j - 1, o - j, j], {j, Min[t, o]}]];
a[n_] :=  b[0, 2n, n] - If[n == 0, 0, b[0, 2n, n - 1]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)

a(n) = A291684(2n,n).

A303203 Number of permutations p of [2n+1] such that 0p has a nonincreasing jump sequence beginning with n+1.

Original entry on oeis.org

1, 2, 12, 98, 988, 10872, 129520, 1609176, 20786804, 274792342, 3719296036, 51119572738, 712918642042, 10055531355652, 143287150725298, 2058368140071146, 29796314629826814, 434051906728752164, 6359063811707227298, 93635249115751389952, 1385028309353547034876

Offset: 0

Alois P. Heinz, Apr 19 2018

nonn

An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.

Alois P. Heinz, Table of n, a(n) for n = 0..250

Bisection (odd part) of A303204.

Cf. A291684.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(sort([u-j, o+j-1])[], j), j=1..min(t, u))+
      add(b(sort([u+j-1, o-j])[], j), j=1..min(t, o)))
    end:
a:= n-> b(0, 2*n+1, n+1)-b(0, 2*n+1, n):
seq(a(n), n=0..25);

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1,
     Sum[b[u-j, o+j-1, j], {j, 1, Min[t, u]}] +
     Sum[b[u+j-1, o-j, j], {j, 1, Min[t, o]}]];
a[n_] := b[0, 2n+1, n+1] - b[0, 2n+1, n];
a /@ Range[0, 25] (* Jean-François Alcover, Sep 01 2021, after Alois P. Heinz *)

a(n) = A291684(2n+1,n+1).

A292168 Number of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with two.

Original entry on oeis.org

1, 2, 5, 9, 17, 31, 57, 101, 185, 333, 599, 1089, 1975, 3563, 6505, 11829, 21455, 39257, 71641, 130403, 239193, 437677, 799127, 1468777, 2693853, 4930871, 9079127, 16684737, 30605159, 56441227, 103900161, 190934999, 352606721, 650072239, 1196527319, 2212404279

Offset: 2

Alois P. Heinz, Sep 10 2017

nonn

An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.

			a(2) = 1: 21.
a(3) = 2: 213, 231.
a(4) = 5: 2134, 2314, 2341, 2413, 2431.
a(5) = 9: 21345, 23145, 23415, 23451, 24135, 24153, 24315, 24351, 24531.
a(6) = 17: 213456, 231456, 234156, 234516, 234561, 241356, 241536, 241563, 243156, 243516, 243561, 245316, 245361, 245631, 246315, 246351, 246531.

Alois P. Heinz, Table of n, a(n) for n = 2..3630

Column k=2 of A291684.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1, j), j=1..min(t, u))+
      add(b(u+j-1, o-j, j), j=1..min(t, o)))
    end:
a:= n-> b(0, n, 2)-b(0, n, 1):
seq(a(n), n=2..50);

A292169 Number of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with three.

Original entry on oeis.org

2, 5, 12, 36, 81, 174, 413, 889, 1870, 4031, 8490, 17580, 36647, 75801, 154676, 316873, 646614, 1309277, 2653548, 5358828, 10786403, 21697201, 43539382, 87208388, 174392929, 348359875, 694913277, 1384281163, 2755398784, 5476741024, 10878139055, 21590446589

Offset: 3

Alois P. Heinz, Sep 10 2017

nonn

An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.

			a(3) = 2: 312, 321.
a(4) = 5: 3124, 3142, 3214, 3241, 3421
a(5) = 12: 31245, 31425, 31452, 32145, 32415, 32451, 34215, 34251, 34521, 35214, 35241, 35421.

Alois P. Heinz, Table of n, a(n) for n = 3..3424

Column k=3 of A291684.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1, j), j=1..min(t, u))+
      add(b(u+j-1, o-j, j), j=1..min(t, o)))
    end:
a:= n-> b(0, n, 3)-b(0, n, 2):
seq(a(n), n=3..50);

A292170 Number of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with four.

Original entry on oeis.org

5, 14, 36, 98, 327, 788, 1890, 4523, 11483, 27209, 61983, 139568, 323531, 730494, 1618531, 3543509, 7851556, 17204960, 37331967, 80312122, 173234350, 372183777, 794587802, 1686743710, 3579347697, 7578433425, 15984149737, 33580700124, 70467297970, 147600492302

Offset: 4

Alois P. Heinz, Sep 10 2017

nonn

An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.

			a(4) = 5: 4123, 4132, 4213, 4231, 4321.
a(5) = 14: 41235, 41325, 41352, 41523, 41532, 42135, 42315, 42351, 42513, 42531, 43215, 43251, 43521, 45321.
a(6) = 36: 412356, 413256, 413526, 413562, 413625, 413652, 415236, 415263, 415326, 415362, 415632, 421356, 423156, 423516, 423561, 425136, 425163, 425316, 425361, 425631, 432156, 432516, 432561, 435216, 435261, 435621, 453216, 453261, 453621, 456321, 463125, 463152, 463215, 463251, 463521, 465321.

Alois P. Heinz, Table of n, a(n) for n = 4..3365

Column k=4 of A291684.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1, j), j=1..min(t, u))+
      add(b(u+j-1, o-j, j), j=1..min(t, o)))
    end:
a:= n-> b(0, n, 4)-b(0, n, 3):
seq(a(n), n=4..50);

A292171 Number of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with five.

Original entry on oeis.org

16, 47, 117, 327, 988, 3392, 8739, 21372, 53596, 135791, 362528, 887060, 2117839, 4997836, 11731828, 28229247, 66196942, 152418888, 347010327, 784580873, 1794241712, 4064606075, 9109879761, 20253187230, 44774963928, 99368298849, 219638865759, 482519177252

Offset: 5

Alois P. Heinz, Sep 10 2017

nonn

An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.

			a(5) = 16: 51234, 51324, 51342, 51423, 51432, 52134, 52314, 52341, 52413, 52431, 53124, 53142, 53214, 53241, 53421, 54321.
a(6) = 47: 512346, 513246, 513426, 513462, 513624, ..., 543216, 543261, 543621, 546321, 564321.

Alois P. Heinz, Table of n, a(n) for n = 5..3335

Column k=5 of A291684.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1, j), j=1..min(t, u))+
      add(b(u+j-1, o-j, j), j=1..min(t, o)))
    end:
a:= n-> b(0, n, 5)-b(0, n, 4):
seq(a(n), n=5..50);

A292172 Number of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with six.

Original entry on oeis.org

52, 166, 425, 1116, 3392, 10872, 38795, 102634, 260334, 651704, 1707641, 4477199, 12277412, 30970764, 75239813, 181934538, 440594545, 1063081418, 2625678546, 6286974074, 14803016600, 34534616815, 80129926919, 185059517397, 431845849360, 997735570874

Offset: 6

Alois P. Heinz, Sep 10 2017

nonn

An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.

			a(6) = 52: 612345, 613245, 613425, 613452, 613524, 613542, 614235, 614253, 614325, 614352, 614532, 615234, 615243, 615324, 615342, 615432, 621345, 623145, 623415, 623451, 624135, 624153, 624315, 624351, 624531, 625134, 625143, 625314, 625341, 625431, 631245, 631425, 631452, 632145, 632415, 632451, 634215, 634251, 634521, 635214, 635241, 635421, 642135, 642315, 642351, 642513, 642531, 643215, 643251, 643521, 645321, 654321.

Alois P. Heinz, Table of n, a(n) for n = 6..2000

Column k=6 of A291684.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1, j), j=1..min(t, u))+
      add(b(u+j-1, o-j, j), j=1..min(t, o)))
    end:
a:= n-> b(0, n, 6)-b(0, n, 5):
seq(a(n), n=6..50);

cp's OEIS Frontend

A303697 Number T(n,k) of permutations p of [n] whose difference between sum of up-jumps and sum of down-jumps equals k; triangle T(n,k), n>=0, min(0,1-n)<=k<=max(0,n-1), read by rows.

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A291685 Number of permutations p of [n] such that 0p has a nonincreasing jump sequence.

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A303204 Number of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with ceiling(n/2).

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A291688 Number of permutations p of [2n] such that 0p has a nonincreasing jump sequence beginning with n.

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A303203 Number of permutations p of [2n+1] such that 0p has a nonincreasing jump sequence beginning with n+1.

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A292168 Number of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with two.

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A292169 Number of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with three.

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A292170 Number of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with four.

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