A335517 -id:A335517 - cp's OEIS frontend

A344605 Number of alternating patterns of length n, including pairs (x,x).

1, 1, 3, 6, 22, 102, 562, 3618, 26586, 219798, 2018686, 20393790, 224750298, 2683250082, 34498833434, 475237879950, 6983085189454, 109021986683046, 1802213242949602, 31447143854808378, 577609702827987882, 11139837273501641502, 225075546284489412854

Offset: 0

Gus Wiseman, May 27 2021

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence is alternating (cf. A025047) including pairs (x,x) if there are no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z. These sequences avoid the weak consecutive patterns (1,2,3) and (3,2,1).
An alternating pattern of length > 2 is necessarily an anti-run (A005649).
The version without pairs (x,x) is identical to this sequence except a(2) = 2 instead of 3.

			The a(0) = 1 through a(4) = 22 patterns:
  ()  (1)  (1,1)  (1,2,1)  (1,2,1,2)
           (1,2)  (1,3,2)  (1,2,1,3)
           (2,1)  (2,1,2)  (1,3,1,2)
                  (2,1,3)  (1,3,2,3)
                  (2,3,1)  (1,3,2,4)
                  (3,1,2)  (1,4,2,3)
                           (2,1,2,1)
                           (2,1,3,1)
                           (2,1,3,2)
                           (2,1,4,3)
                           (2,3,1,2)
                           (2,3,1,3)
                           (2,3,1,4)
                           (2,4,1,3)
                           (3,1,2,1)
                           (3,1,3,2)
                           (3,1,4,2)
                           (3,2,3,1)
                           (3,2,4,1)
                           (3,4,1,2)
                           (4,1,3,2)
                           (4,2,3,1)

Martin Ehrenstein, Table of n, a(n) for n = 0..25

The version for permutations is A001250.
The version for compositions is A344604.
The version for permutations of prime indices is A344606.
A000670 counts patterns (ranked by A333217).
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A019536 counts necklace patterns.
A025047 counts alternating or wiggly compositions, complement A345192.
A226316 counts patterns avoiding (1,2,3) (weakly: A052709).
A335515 counts patterns matching (1,2,3).
Cf. A000041, A006330, A049774, A102726, A103919, A124754, A128761, A335456, A335457, A335517, A344612, A344614, A344615, A348377.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]],{n,0,6}]

a(10) and beyond from Martin Ehrenstein, Jun 10 2021

A345194 Number of alternating patterns of length n.

Original entry on oeis.org

1, 1, 2, 6, 22, 102, 562, 3618, 26586, 219798, 2018686, 20393790, 224750298, 2683250082, 34498833434, 475237879950, 6983085189454, 109021986683046, 1802213242949602, 31447143854808378, 577609702827987882, 11139837273501641502, 225075546284489412854

Offset: 0

Gus Wiseman, Jun 17 2021

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). An alternating pattern is necessarily an anti-run (A005649).
The version with twins (A344605) is identical to this sequence except with a(2) = 3 instead of 2.
From Gus Wiseman, Jan 16 2022: (Start)
Conjecture: Also the number of weakly up/down patterns of length n, where a sequence is weakly up/down if it is alternately weakly increasing and weakly decreasing, starting with an increase. For example, the a(0) = 1 through a(3) = 6 weakly up/down patterns are:
  ()  (1)  (1,1)  (1,1,1)
           (2,1)  (1,1,2)
                  (2,1,1)
                  (2,1,2)
                  (2,1,3)
                  (3,1,2)
(End)

			The a(0) = 1 through a(3) = 6 alternating patterns:
  ()  (1)  (1,2)  (1,2,1)
           (2,1)  (1,3,2)
                  (2,1,2)
                  (2,1,3)
                  (2,3,1)
                  (3,1,2)

Andrew Howroyd, Table of n, a(n) for n = 0..200

The version for permutations is A001250, complement A348615.
The version for compositions is A025047, complement A345192.
The version with twins (x,x) is A344605.
The version for perms of prime indices is A345164, complement A350251.
The version for factorizations is A348610, complement A348613, weak A349059.
The weak version is A349058, complement A350138, compositions A349052.
The complement is counted by A350252.
A000670 = patterns, ranked by A333217.
A003242 = anti-run compositions.
A005649 = anti-run patterns, complement A069321.
A019536 = necklace patterns.
A129852 and A129853 = up/down and down/up compositions.
A226316 = patterns avoiding (1,2,3), weakly A052709, complement A335515.
A345170 = partitions w/ alternating permutation, complement A345165.
A349055 = normal multisets w/ alternating permutation, complement A349050.
Cf. A006330, A049774, A128761, A335456, A335457, A335517, A344614, A344615, A345167, A345168.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
allnorm[n_]:=If[n<=0,{{}},Function[s, Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],wigQ]],{n,0,6}]

F(p,x) = {sum(k=0, p, (-1)^((k+1)\2)*binomial((p+k)\2, k)*x^k)}
R(n,k) = {Vec(if(k==1, x, 2*F(k-2,-x)/F(k-1,x)-2-(k-2)*x) + O(x*x^n))}
seq(n)= {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Feb 04 2022

a(n) = 2*A350354(n) for n >= 2. - Andrew Howroyd, Feb 04 2022

a(10)-a(18) from Alois P. Heinz, Dec 10 2021

Terms a(19) and beyond from Andrew Howroyd, Feb 04 2022

A019472 Weak preference orderings of n alternatives, i.e., orderings that have indifference between at least two alternatives.

Original entry on oeis.org

0, 0, 1, 7, 51, 421, 3963, 42253, 505515, 6724381, 98618763, 1582715773, 27612565995, 520631327581, 10554164679243, 228975516609853, 5294731892093355, 130015079601039901, 3379132289551117323, 92679942218919579133, 2675254894236207563115, 81073734056332364441821

Offset: 0

Robert Ware (bware(AT)wam.umd.edu)

From Gus Wiseman, Jun 24 2020: (Start)
Equivalently, a(n) is number of (1,1)-matching sequences of length n that cover an initial interval of positive integers. For example, the a(2) = 1 and a(3) = 7 sequences are:
  (1,1)  (1,1,1)
         (1,1,2)
         (1,2,1)
         (1,2,2)
         (2,1,1)
         (2,1,2)
         (2,2,1)
Missing from this list are:
  (1,2)  (1,2,3)
  (2,1)  (1,3,2)
         (2,1,3)
         (2,3,1)
         (3,1,2)
         (3,2,1)
(End)

Wikipedia, Weak ordering
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Cf. A000670, A052875.
(1,1)-avoiding patterns are counted by A000142.
(1,2)-matching patterns are counted by A056823.
(1,1)-matching compositions are counted by A261982.
(1,1)-matching compositions are ranked by A335488.
Patterns matched by patterns are counted by A335517.
Cf. A056986, A333217, A335454, A335456, A335515.

a[n_] := Sum[(-1)^(j-i)*Binomial[j, i]*i^n, {i, 0, n-1}, {j, 0, n-1}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Feb 26 2016, after Peter Luschny *)

def A019472(n):
    return add(add((-1)^(j-i)*binomial(j, i)*i^n for i in range(n)) for j in range(n))
[A019472(n) for n in range(21)] # Peter Luschny, Jul 22 2014

a(n) = A000670(n) - n!. - corrected by Eugene McDonnell, May 12 2000

a(n) = Sum_{j=0..n-1} Sum_{i=0..n-1} (-1)^(j-i)*C(j, i)*i^n. - Peter Luschny, Jul 22 2014

A335518 Number of matching pairs of patterns, the first of length n and the second of length k.

Original entry on oeis.org

1, 1, 1, 3, 3, 3, 13, 13, 25, 13, 75, 75, 185, 213, 75, 541, 541, 1471, 2719, 2053, 541, 4683, 4683, 13265, 32973, 40367, 22313, 4683, 47293, 47293, 136711, 408265, 713277, 625295, 271609, 47293

Offset: 0

Gus Wiseman, Jun 23 2020

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			Triangle begins:
     1
     1     1
     3     3     3
    13    13    25    13
    75    75   185   213    75
   541   541  1471  2719  2053   541
  4683  4683 13265 32973 40367 22313  4683
Row n =2 counts the following pairs:
  ()<=(1,1)  (1)<=(1,1)  (1,1)<=(1,1)
  ()<=(1,2)  (1)<=(1,2)  (1,2)<=(1,2)
  ()<=(2,1)  (1)<=(2,1)  (2,1)<=(2,1)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Columns k = 0 and k = 1 are both A000670.
Row sums are A335517.
Patterns are ranked by A333217.
Patterns matched by a standard composition are counted by A335454.
Patterns contiguously matched by compositions are counted by A335457.
Minimal patterns avoided by a standard composition are counted by A335465.
Patterns matched by prime indices are counted by A335549.
Cf. A011782, A034691, A056986, A124771, A269134, A329744, A333257, A334299.

mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Sum[Length[Union[mstype/@Subsets[y,{k}]]],{y,Join@@Permutations/@allnorm[n]}],{n,0,5},{k,0,n}]

cp's OEIS Frontend

A344605 Number of alternating patterns of length n, including pairs (x,x).

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A345194 Number of alternating patterns of length n.

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A019472 Weak preference orderings of n alternatives, i.e., orderings that have indifference between at least two alternatives.

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A335518 Number of matching pairs of patterns, the first of length n and the second of length k.

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