A056986 -id:A056986 - cp's OEIS frontend

A335837 Number of normal patterns matched by integer partitions of n.

1, 2, 5, 9, 18, 31, 54, 89, 146, 228, 358, 545, 821, 1219, 1795, 2596, 3741, 5323, 7521, 10534, 14659, 20232, 27788, 37897, 51410, 69347, 93111, 124348, 165378, 218924, 288646, 379021, 495864, 646272, 839490, 1086693, 1402268, 1803786, 2313498, 2958530, 3773093

Offset: 0

Gus Wiseman, Jun 27 2020

nonn

We define a (normal) 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).

			The a(0) = 1 through a(4) = 18  pairs of a partition with a matched pattern:
  ()/()  (1)/()   (2)/()     (3)/()       (4)/()
         (1)/(1)  (2)/(1)    (3)/(1)      (4)/(1)
                  (11)/()    (21)/()      (31)/()
                  (11)/(1)   (21)/(1)     (31)/(1)
                  (11)/(11)  (21)/(21)    (31)/(21)
                             (111)/()     (22)/()
                             (111)/(1)    (22)/(1)
                             (111)/(11)   (22)/(11)
                             (111)/(111)  (211)/()
                                          (211)/(1)
                                          (211)/(11)
                                          (211)/(21)
                                          (211)/(211)
                                          (1111)/()
                                          (1111)/(1)
                                          (1111)/(11)
                                          (1111)/(111)
                                          (1111)/(1111)

Christian Sievers, Table of n, a(n) for n = 0..2000
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for compositions in standard order is A335454.
The version for compositions is A335456.
The version for Heinz numbers of partitions is A335549.
The contiguous case is A335838.
Patterns are counted by A000670 and ranked by A333217.
Patterns contiguously matched by prime indices are A335516.
Contiguous divisors are counted by A335519.
Minimal patterns avoided by prime indices are counted by A335550.
Cf. A000005, A056986, A108917, A269134, A333257, A334299, A335458, A335465.

mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Sum[Length[Union[mstype/@Subsets[y]]],{y,IntegerPartitions[n]}],{n,0,8}]

lista(n) = {
  my(v=vector(n+1,i,1+x*O(x^n)));
  for(k=1,n,
    v=vector(n\(k+1)+1,i,
        (1-x^(i*k))/(1-x^k)*v[i] + sum(j=i,n\k,x^(j*k)*v[j+1]) +
        x^(k*i)/(1-x^k)^2*v[1] ) );
  Vec(v[1]) } \\ Christian Sievers, May 08 2025

a(18) corrected by and a(19)-a(22) from Jinyuan Wang, Jun 27 2020

More terms from Christian Sievers, May 08 2025

A356116 Triangle read by row. The reduced triangle of the partition_triangle A355776.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 5, 5, 0, 0, 16, 46, 16, 0, 0, 42, 252, 252, 42, 0, 0, 99, 1086, 2241, 1086, 99, 0, 0, 219, 4097, 15129, 15129, 4097, 219, 0, 0, 466, 14272, 87058, 154426, 87058, 14272, 466, 0, 0, 968, 47300, 452672, 1305062, 1305062, 452672, 47300, 968, 0

Offset: 1

Peter Luschny, Jul 28 2022

By a partition triangle, we understand an irregular triangle where each row corresponds to a mapping of Partitions(n) -> ZZ. We assume a fixed order of the partitions given. Here we will use the ordering defined in A080577. Examples are A355776, A355777, and A134264. 'Reducing' then means summing the values corresponding to the partitions of n with length k. The 'reduced partition triangle' then is a regular triangle with T(n, k) with 1 <= k <= n.

Conversely, A355776, the statistic of permutations whose Lehmer code is nonmonotonic, can be seen as a refinement of this triangle, which in turn is a refinement of the sequence A056986, the number of permutations on [n] containing any given pattern alpha in the symmetric group S_3.

			Triangle T(n, k) starts:
[1] [0]
[2] [0,   0]
[3] [0,   1,     0]
[4] [0,   5,     5,      0]
[5] [0,  16,    46,     16,       0]
[6] [0,  42,   252,    252,      42,       0]
[7] [0,  99,  1086,   2241,    1086,      99,      0]
[8] [0, 219,  4097,  15129,   15129,    4097,    219,     0]
[9] [0, 466, 14272,  87058,  154426,   87058,  14272,   466,   0]
[10][0, 968, 47300, 452672, 1305062, 1305062, 452672, 47300, 968, 0]
.
Row 6 of the partition triangle A355776 is:
[0, [10, 20, 12], [61,  162, 29], [102, 150], 42, 0]
Adding the bracketed terms reduces this row to row 6 of the above triangle.

FindStat - Combinatorial Statistic Finder, The number of occurrences of the pattern [1,2,3] inside a permutation of length at least 3.
Peter Luschny, Permutations with Lehmer, a SageMath Jupyter Notebook.

A002662 (column 1), A056986 (row sums), A355776 (refinement).

from functools import cache
@cache
def Pn(n: int, k: int) -> int:
    if k == 0: return 0
    if n == 0 or k == 1: return 1
    return Pn(n, k - 1) + Pn(n - k, k) if k <= n else Pn(n, k - 1)
def reduce_parts(fun, n: int) -> list[int]:
    funn: list[int] = fun(n)
    return [sum(funn[Pn(n, k):Pn(n, k + 1)]) for k in range(n)]
def reduce_partition_triangle(fun, n: int) -> list[list[int]]:
    return [reduce_parts(fun, k) for k in range(1, n)]
reduce_partition_triangle(A355776_row, 6)

A158432 Number of permutations of 1..n containing the relative rank sequence { 45312 } at any spacing.

Original entry on oeis.org

1, 26, 458, 6996, 101072, 1438112, 20598112, 300892896, 4521034917, 70286670034, 1135485759114, 19121776482564, 336412530327804, 6191800556586104, 119301546930406184, 2406376964044265344, 50786085223779295344, 1120447461653440780128, 25810064637612342838624

Offset: 5

R. H. Hardin, Mar 18 2009

nonn

Same series for 54321 12345 45321 21345 12354 54312 34521 32145 12543 54123 23451 43215 15432 51234 21354 34512 32154 21543 45123.

Alois P. Heinz, Table of n, a(n) for n = 5..200
Eric Weisstein's World of Mathematics, Permutation Pattern

Cf. A000142, A047889, A056986, A158005, A214015, A214152.

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
      +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
                 add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
a:= n-> n! -g(n, 4, []):
seq(a(n), n=5..25);  # Alois P. Heinz, Jul 05 2012
# second Maple program
a:= proc(n) option remember; `if`(n<5, 0, `if`(n=5, 1,
     ((132-142*n-301*n^2-35*n^3+25*n^4+n^5)*a(n-1)
     -2*(10*n^3+33*n^2-181*n-2)*(n-1)^2*a(n-2)
     +64*(n-2)^2*(n-1)^3*a(n-3))/ ((n+4)*(n-5)*(n+3)^2)))
    end:
seq(a(n), n=5..30);  # Alois P. Heinz, Sep 26 2012

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1+l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n == 0 || i === 1, h[Join[l, Array[1 &, n]]]^2, If[i < 1, 0, Sum[g[n - i*j, i - 1, Join[l, Array[i &, j]]], {j, 0, n/i}]]];
a[n_] := n! - g[n, 4, {}];
Table[a[n], {n, 5, 25}] (* Jean-François Alcover, Jun 19 2018, after Alois P. Heinz's first program *)

a(n) = A214152(n,5) = A000142(n)-A047889(n) = A000142(n)-A214015(n,4).

Extended beyond a(16) by Alois P. Heinz, Jul 05 2012

A159139 Number of permutations of 1..n containing the relative rank sequence { 213465 } at any spacing.

Original entry on oeis.org

1, 37, 891, 18043, 337210, 6081686, 108469917, 1941309261, 35187952132, 649951312000, 12286366975723, 238445927000811, 4762398793018878, 98074791689121162, 2085684931155975120, 45859509146309390064, 1043533983233372354613, 24590543663448304800169

Offset: 6

R. H. Hardin, Apr 05 2009

nonn

Same series for 654321 123456 564321 213456 123465 654312 456321 321456 123654 654123 345621 432156 126543 651234 564312 456312 321465 213654 564123 345612 432165 216543 561234 234561 543216 165432 612345 456123 321654.

Alois P. Heinz, Table of n, a(n) for n = 6..200
Eric Weisstein's World of Mathematics, Permutation Pattern

Cf. A000142, A047890, A056986, A158005, A158423, A214015, A214152.

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
      +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l)
      `if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
       add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> n! -g(n, 5, []):
seq(a(n), n=6..30);  # Alois P. Heinz, Jul 05 2012
# second Maple program
a:= proc(n) option remember; `if`(n<6, 0, `if`(n=6, 1,
     ((2475-4819*n^2-2985*n+175*n^4-1021*n^3+n^6+49*n^5)*a(n-1)
     -(35*n^4+441*n^3-845*n^2-4147*n-489)*(n-1)^2*a(n-2)
     +(-1668+329*n+259*n^2)*(n-1)^2*(n-2)^2*a(n-3)
     -225*(n-1)^2*(n-2)^2*(n-3)^2*a(n-4))/ ((n-6)*(n+6)^2*(n+4)^2)))
    end:
seq(a(n), n=6..30);  # Alois P. Heinz, Sep 26 2012

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1+l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n == 0 || i === 1, h[Join[l, Array[1 &, n]]]^2, If[i < 1, 0, Sum[g[n - i*j, i - 1, Join[l, Array[i &, j]]], {j, 0, n/i}]]];
a[n_] := n! - g[n, 5, {}];
Table[a[n], {n, 6, 30}] (* Jean-François Alcover, Jun 19 2018, from first Maple program *)

a(n) = A214152(n,6) = A000142(n)-A047890(n) = A000142(n)-A214015(n,5). - Alois P. Heinz, Jul 05 2012

More terms from Alois P. Heinz, Jul 05 2012

A159175 Number of permutations of 1..n containing the relative rank sequence { 1234567 } at any spacing.

Original entry on oeis.org

1, 50, 1578, 40884, 958809, 21353634, 463945294, 9996042284, 215831724525, 4702905606350, 103912444955422, 2336099774748540, 53567906041439136, 1255172323669315848, 30095426182382305848, 739238316780966277616, 18619024923770934306358, 481234428294016650524172

Offset: 7

R. H. Hardin Apr 05 2009

nonn

Same series (among rank sequences with inversion = reversal) for 3214765 2134576.

Alois P. Heinz, Table of n, a(n) for n = 7..200
Eric Weisstein's World of Mathematics, Permutation Pattern

Cf. A000142, A052399, A056986, A158005, A158423, A214015, A214152.

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
      +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
                 add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
a:= n-> n! -g(n, 6, []):
seq(a(n), n=7..25);  # Alois P. Heinz, Jul 05 2012
# second Maple program
a:= proc(n) option remember; `if`(n<7, 0, `if`(n=7, 1, ((-93464*n+1072*n^4
      +72128-125284*n^2+84*n^6+994*n^5-30491*n^3+n^7) *a(n-1)
      -4*(14*n^5+399*n^4+1124*n^3-7354*n^2-23983*n-5042)*(n-1)^2 *a(n-2)
      +4*(-7359-2629*n+1596*n^2+196*n^3)*(n-1)^2*(n-2)^2 *a(n-3)
      -1152*(1+2*n)*(n-1)^2*(n-2)^2*(n-3)^2 *a(n-4))/
       ((n-7)*(n+9)*(n+8)^2*(n+5)^2)))
    end:
seq(a(n), n=7..30);  # Alois P. Heinz, Sep 27 2012

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1+l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n == 0 || i === 1, h[Join[l, Array[1 &, n]]]^2, If[i < 1, 0, Sum[g[n - i*j, i - 1, Join[l, Array[i &, j]]], {j, 0, n/i}]]];
a[n_] := n! - g[n, 6, {}];
Table[a[n], {n, 7, 25}] (* Jean-François Alcover, Jun 19 2018, from first Maple program *)

a(n) = A214152(n,7) = A000142(n)-A052399(n) = A000142(n)-A214015(n,6). - Alois P. Heinz, Jul 05 2012

Extended beyond a(16) by Alois P. Heinz, Jul 05 2012

A335447 Number of (1,2)-matching permutations of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 5, 0, 0, 1, 1, 1, 5, 0, 1, 1, 3, 0, 5, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 11, 0, 1, 2, 0, 1, 5, 0, 2, 1, 5, 0, 9, 0, 1, 2, 2, 1, 5, 0, 4, 0, 1, 0, 11, 1, 1

Offset: 1

Gus Wiseman, Jun 14 2020

nonn

Depends only on sorted prime signature (A118914).
Also the number of (2,1)-matching permutations of the prime indices of n.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. 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).

			The a(n) permutations for n = 6, 12, 24, 48, 30, 72, 60:
  (12)  (112)  (1112)  (11112)  (123)  (11122)  (1123)
        (121)  (1121)  (11121)  (132)  (11212)  (1132)
               (1211)  (11211)  (213)  (11221)  (1213)
                       (12111)  (231)  (12112)  (1231)
                                (312)  (12121)  (1312)
                                       (12211)  (1321)
                                       (21112)  (2113)
                                       (21121)  (2131)
                                       (21211)  (2311)
                                                (3112)
                                                (3121)

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

The avoiding version is A000012.
Patterns are counted by A000670.
Positions of zeros are A000961.
(1,2)-matching patterns are counted by A002051.
Permutations of prime indices are counted by A008480.
(1,2)-matching compositions are counted by A056823.
STC-numbers of permutations of prime indices are A333221.
Patterns matched by standard compositions are counted by A335454.
(1,2,1) or (2,1,2)-matching permutations of prime indices are A335460.
(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
Dimensions of downsets of standard compositions are A335465.
(1,2)-matching compositions are ranked by A335485.
Cf. A056239, A056986, A112798, A181796, A333175, A335451, A335452, A335463, A333175.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],!GreaterEqual@@#&]],{n,100}]

a(n) = A008480(n) - 1.

A335476 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,1,2).

Original entry on oeis.org

14, 28, 29, 30, 46, 54, 56, 57, 58, 59, 60, 61, 62, 78, 84, 92, 93, 94, 102, 108, 109, 110, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 142, 156, 157, 158, 168, 169, 172, 174, 180, 182, 184, 185, 186, 187, 188, 189, 190, 198, 204

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. 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.

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).

			The sequence of terms together with the corresponding compositions begins:
  14: (1,1,2)
  28: (1,1,3)
  29: (1,1,2,1)
  30: (1,1,1,2)
  46: (2,1,1,2)
  54: (1,2,1,2)
  56: (1,1,4)
  57: (1,1,3,1)
  58: (1,1,2,2)
  59: (1,1,2,1,1)
  60: (1,1,1,3)
  61: (1,1,1,2,1)
  62: (1,1,1,1,2)
  78: (3,1,1,2)
  84: (2,2,3)

Wikipedia, Permutation pattern
Gus Wiseman, Statistics, classes, and transformations of standard compositions
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The complement A335522 is the avoiding version.
The (2,1,1)-matching version is A335478.
Patterns matching this pattern are counted by A335509 (by length).
Permutations of prime indices matching this pattern are counted by A335446.
These compositions are counted by A335470 (by sum).
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,x_,_,y_,_}/;x

A335477 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,2,1).

Original entry on oeis.org

21, 43, 45, 53, 73, 85, 86, 87, 91, 93, 107, 109, 117, 146, 147, 149, 153, 165, 169, 171, 172, 173, 174, 175, 181, 182, 183, 187, 189, 201, 213, 214, 215, 219, 221, 235, 237, 245, 273, 277, 293, 294, 295, 297, 299, 301, 306, 307, 309, 313, 325, 329, 331, 333

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. 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.

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).

			The sequence of terms together with the corresponding compositions begins:
   21: (2,2,1)
   43: (2,2,1,1)
   45: (2,1,2,1)
   53: (1,2,2,1)
   73: (3,3,1)
   85: (2,2,2,1)
   86: (2,2,1,2)
   87: (2,2,1,1,1)
   91: (2,1,2,1,1)
   93: (2,1,1,2,1)
  107: (1,2,2,1,1)
  109: (1,2,1,2,1)
  117: (1,1,2,2,1)
  146: (3,3,2)
  147: (3,3,1,1)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The complement A335524 is the avoiding version.
The (1,2,2)-matching version is A335475.
Patterns matching this pattern are counted by A335509 (by length).
Permutations of prime indices matching this pattern are counted by A335453.
These compositions are counted by A335472 (by sum).
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,x_,_,y_,_}/;x>y]&]

A355776 Partition triangle read by rows. A statistic of permutations whose Lehmer code is nonmonotonic, refining triangle A356116.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 3, 2, 5, 0, 0, 6, 10, 22, 24, 16, 0, 0, 10, 20, 12, 61, 162, 29, 102, 150, 42, 0, 0, 15, 35, 49, 135, 432, 246, 273, 391, 1389, 461, 388, 698, 99, 0, 0, 21, 56, 90, 52, 260, 982, 1288, 740, 827, 1150, 4974, 2745, 5778, 482, 2057, 8924, 4148, 1333, 2764, 219, 0

Offset: 0

Peter Luschny, Jul 27 2022

We say a list L is weakly increasing if x <= y, and weakly decreasing, if x >= y, for all x, y in L if index(x) < index(y). We say a list L is nonmonotonic if it is not weakly increasing and not weakly decreasing.

The ordering of the partitions is defined in A334439. See the comments in A356116 for the definition of the terms 'partition triangle' and 'reduced partition triangle'.

			Table T(n, k) starts:
[0]  0;
[1]  0;
[2]  0,  0;
[3]  0,  1,  0;
[4]  0, [3,  2],   5,  0;
[5]  0, [6, 10], [22, 24],   16,  0;
[6]  0, [10, 20, 12], [61,  162, 29], [102, 150],   42,   0;
[7]  0, [15, 35, 49], [135, 432, 246, 273], [391, 1389, 461], [388, 698], 99, 0;
Summing the bracketed terms reduces the triangle to A356116.
.
The permutations whose Lehmer code is nonmonotonic, in the case n = 4, k = 1 are: 1243, 1324, 1423, which map to the partition [3, 1] and 1342, 2143, which map to the partition [2, 2]. Thus A356116(4, 1) = 3 + 2 = 5.
.
The cardinality of the preimage of the partitions, i.e. the number of permutations whose Lehmer code is nonmonotonic, are the terms of the sequence. Here row 6:
[6] => 0
[5, 1] => 10
[4, 2] => 20
[3, 3] => 12
[4, 1, 1] => 61
[3, 2, 1] => 162
[2, 2, 2] => 29
[3, 1, 1, 1] => 102
[2, 2, 1, 1] => 150
[2, 1, 1, 1, 1] => 42
[1, 1, 1, 1, 1, 1] => 0

Peter Luschny, Permutations with Lehmer, a SageMath Jupyter Notebook.

Cf. A000217 (column 1), A002662 (subdiagonal), A000041 (row lengths), A056986 (row sums), A356116 (reduced triangle), A355777 (Euler-Lehmer).

import collections
def perm_lehmer_nonmono_stats(n):
    res = collections.defaultdict(int)
    for p in Permutations(n):
        l = p.to_lehmer_code()
        if all(x >= y for x, y in zip(l, l[1:])): continue
        c = [l.count(i) for i in range(len(p)) if i in l]
        res[Partition(reversed(sorted(c)))] += 1
    return sorted(res.items(), key=lambda x: len(x[0]))
@cached_function
def A355776_row(n):
    if n < 2: return [0]
    S = perm_lehmer_nonmono_stats(n)
    return [0] + [s[1] for s in S] + [0]
def A355776(n, k): return A355776_row(n)[k] if n > 0 else 0
for n in range(0, 8): print(A355776_row(n))

A335478 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,1,1).

Original entry on oeis.org

11, 19, 23, 27, 35, 39, 43, 45, 46, 47, 51, 55, 59, 67, 71, 74, 75, 77, 78, 79, 83, 87, 89, 91, 92, 93, 94, 95, 99, 103, 107, 109, 110, 111, 115, 119, 123, 131, 135, 138, 139, 141, 142, 143, 147, 149, 150, 151, 153, 154, 155, 156, 157, 158, 159, 163, 167, 171

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. 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.

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).

			The sequence of terms together with the corresponding compositions begins:
  11: (2,1,1)
  19: (3,1,1)
  23: (2,1,1,1)
  27: (1,2,1,1)
  35: (4,1,1)
  39: (3,1,1,1)
  43: (2,2,1,1)
  45: (2,1,2,1)
  46: (2,1,1,2)
  47: (2,1,1,1,1)
  51: (1,3,1,1)
  55: (1,2,1,1,1)
  59: (1,1,2,1,1)
  67: (5,1,1)
  71: (4,1,1,1)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The complement A335523 is the avoiding version.
The (1,1,2)-matching version is A335476.
Patterns matching this pattern are counted by A335509 (by length).
Permutations of prime indices matching this pattern are counted by A335516.
These compositions are counted by A335470 (by sum).
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458, A335475.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,y_,_,y_,_}/;x>y]&]

cp's OEIS Frontend

A335837 Number of normal patterns matched by integer partitions of n.

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A356116 Triangle read by row. The reduced triangle of the partition_triangle A355776.

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A158432 Number of permutations of 1..n containing the relative rank sequence { 45312 } at any spacing.

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A159139 Number of permutations of 1..n containing the relative rank sequence { 213465 } at any spacing.

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A159175 Number of permutations of 1..n containing the relative rank sequence { 1234567 } at any spacing.

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A335447 Number of (1,2)-matching permutations of the prime indices of n.

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A335476 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,1,2).

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