A005649 -id:A005649 - cp's OEIS frontend

A226515 Row 2 of array in A226513.

1, 3, 15, 99, 807, 7803, 87135, 1102419, 15575127, 242943723, 4145495055, 76797289539, 1534762643847, 32907617073243, 753473367606975, 18347287182129459, 473409784213526967, 12902366605394652363, 370357953441110390895, 11167936445234485414179

Offset: 0

N. J. A. Sloane, Jun 13 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.

Cf. rows 0, 1, 3, 4, 5 of A226513: A000670, A005649, A226738, A226739, A226740.

m:=2; [&+[StirlingSecond(n, i)*Factorial(i)*Binomial(m+i, i): i in [0..n]]: n in [0..20]]; // Bruno Berselli, Jun 18 2013

Range[0, 20]! CoefficientList[Series[(2 - Exp@x)^-3, {x, 0, 20}], x] (* Vincenzo Librandi, Jun 18 2013 *)

E.g.f.: 1/(2 - exp(x))^3 (see the Ahlbach et al. paper, Theorem 4). - Vincenzo Librandi, Jun 18 2013
a(n) = Sum_{i=0..n} S2(n,i)*i!*binomial(2+i,i), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3). [Bruno Berselli, Jun 18 2013]
G.f.: 1/Q(0), where Q(k) = 1 - 3*x*(k + 1) - 2*x^2*(k + 1)*(k + 3)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 02 2013
G.f.: 1/(1 + x)/Q(0,u), where u = x/(1 + x), Q(k,u) = 1 - u*(3*k + 4) - 2*u^2*(k + 1)*(k + 3)/Q(k+1,u); (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013
a(n) ~ n! * n^2 /(16*(log(2))^(n + 3)) * (1 + 3*(1 + log(2))/n). - Vaclav Kotesovec, Oct 08 2013
Conjectural g.f. as a continued fraction of Stieltjes type: 1/(1 - 3*x/(1 - 2*x/(1 - 4*x/(1 - 4*x/(1 - 5*x/(1 - 6*x/(1 - (n+2)*x/(1 - 2*n*x/(1 - ... ))))))))). - Peter Bala, Aug 27 2023
From Seiichi Manyama, Nov 19 2023: (Start)
a(0) = 1; a(n) = Sum_{k=1..n} (2*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 3*a(n-1) - 2*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). (End)

More terms from Vincenzo Librandi, Jun 18 2013

A386634 Number of inseparable type set partitions of {1..n}.

Original entry on oeis.org

0, 0, 1, 1, 5, 6, 37, 50, 345, 502, 3851, 5897, 49854, 79249, 730745, 1195147, 11915997, 19929390, 213332101, 363275555, 4150104224, 7172334477, 87003759195, 152231458128, 1952292972199, 3451893361661, 46625594567852, 83183249675125, 1179506183956655, 2120758970878892

Offset: 0

Gus Wiseman, Aug 09 2025

nonn

A set partition is of inseparable type iff the underlying set has no permutation whose adjacent elements always belong to different blocks. Note that this only depends on the sizes of the blocks.
A set partition is also of inseparable type iff its greatest block size is at least 2 more than the sum of its other block sizes.
This is different from inseparable partitions (A325535) and partitions of inseparable type (A386638 or A025065).

			The a(2) = 1 through a(5) = 6 set partitions:
  {{1,2}}  {{1,2,3}}  {{1,2,3,4}}    {{1,2,3,4,5}}
                      {{1},{2,3,4}}  {{1},{2,3,4,5}}
                      {{1,2,3},{4}}  {{1,2,3,4},{5}}
                      {{1,2,4},{3}}  {{1,2,3,5},{4}}
                      {{1,3,4},{2}}  {{1,2,4,5},{3}}
                                     {{1,3,4,5},{2}}

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

For separable partitions we have A386583, sums A325534, ranks A335433.
For inseparable partitions we have A386584, sums A325535, ranks A335448.
For separable type partitions we have A386585, sums A336106, ranks A335127.
For inseparable type partitions we have A386586, sums A386638 or A025065, ranks A335126.
The complement is counted by A386633, sums of A386635.
Row sums of A386636.
A000110 counts set partitions, row sums of A048993.
A000670 counts ordered set partitions.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A279790 counts disjoint families on strongly normal multisets.
A335434 counts separable factorizations, inseparable A333487.
A336103 counts normal separable multisets, inseparable A336102.
A386587 counts disjoint families of strict partitions of each prime exponent.
Cf. A001055, A124762, A239455, A335125, A386575, A386579.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
stnseps[stn_]:=Select[Permutations[Union@@stn],And@@Table[Position[stn,#[[i]]][[1,1]]!=Position[stn,#[[i+1]]][[1,1]],{i,Length[#]-1}]&]
Table[Length[Select[sps[Range[n]],stnseps[#]=={}&]],{n,0,5}]

a(12)-a(29) from Alois P. Heinz, Aug 10 2025

A335407 Number of anti-run permutations of the prime indices of n!.

Original entry on oeis.org

1, 1, 1, 2, 0, 2, 3, 54, 0, 30, 105, 6090, 1512, 133056, 816480, 127209600, 0, 10090080, 562161600, 69864795000, 49989139200, 29593652088000, 382147120555200, 41810689605484800, 4359985823793600, 3025062801079038720, 49052072750637116160, 25835971971637227375360

Offset: 0

Gus Wiseman, Jul 01 2020

nonn

An anti-run is a sequence with no adjacent equal parts.
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.
Conjecture: Only vanishes at n = 4 and n = 8.
a(16) = 0. Proof: 16! = 2^15 * m where bigomega(m) = A001222(m) = 13. We can't separate 15 1's with 13 other numbers. - David A. Corneth, Jul 04 2020

			The a(0) = 1 through a(6) = 3 anti-run permutations:
  ()  ()  (1)  (1,2)  .  (1,2,1,3,1)  (1,2,1,2,1,3,1)
               (2,1)     (1,3,1,2,1)  (1,2,1,3,1,2,1)
                                      (1,3,1,2,1,2,1)

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

The version for Mersenne numbers is A335432.
Anti-run compositions are A003242.
Anti-run patterns are counted by A005649.
Permutations of prime indices are A008480.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Strict permutations of prime indices are A335489.
Cf. A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335125.
Factorial numbers: A000142, A001222, A002982, A007489, A022559, A027423, A054991, A108731, A181819, A181821, A325272, A325273, A325617.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n!]],!MatchQ[#,{_,x_,x_,_}]&]],{n,0,10}]

\\ See A335452 for count.
a(n)={count(factor(n!)[,2])} \\ Andrew Howroyd, Feb 03 2021

a(n) = A335452(A000142(n)). - Andrew Howroyd, Feb 03 2021

Terms a(14) and beyond from Andrew Howroyd, Feb 03 2021

A350252 Number of non-alternating patterns of length n.

Original entry on oeis.org

0, 0, 1, 7, 53, 439, 4121, 43675, 519249, 6867463, 100228877, 1602238783, 27866817297, 524175098299, 10606844137009, 229807953097903, 5308671596791901, 130261745042452855, 3383732450013895721, 92770140175473602755, 2677110186541556215233

Offset: 0

Gus Wiseman, Jan 13 2022

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).
Conjecture: Also the number of non-weakly up/down (or down/up) patterns of length n. For example:
- The a(3) = 7 non-weakly up/down patterns:
  (121), (122), (123), (132), (221), (231), (321)
- The a(3) = 7 non-weakly down/up patterns:
  (112), (123), (211), (212), (213), (312), (321)
- The a(3) = 7 non-alternating patterns (see example for more):
  (111), (112), (122), (123), (211), (221), (321)

			The a(2) = 1 and a(3) = 7 non-alternating patterns:
  (1,1)  (1,1,1)
         (1,1,2)
         (1,2,2)
         (1,2,3)
         (2,1,1)
         (2,2,1)
         (3,2,1)
The a(4) = 53 non-alternating patterns:
  2112   3124   4123   1112   2134   1234   3112   2113   1123
  2211   3214   4213   1211   2314   1243   3123   2123   1213
  2212   3412   4312   1212   2341   1324   3211   2213   1223
         3421   4321   1221   2413   1342   3212   2311   1231
                       1222   2431   1423   3213   2312   1232
                                     1432   3312   2313   1233
                                            3321   2321   1312
                                                   2331   1321
                                                          1322
                                                          1323
                                                          1332

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

The unordered version is A122746.
The version for compositions is A345192, ranked by A345168, weak A349053.
The complement is counted by A345194, weak A349058.
The version for factorizations is A348613, complement A348610, weak A350139.
The strict case (permutations) is A348615, complement A001250.
The weak version for partitions is A349061, complement A349060.
The weak version for perms of prime indices is A349797, complement A349056.
The weak version is A350138.
The version for perms of prime indices is A350251, complement A345164.
A000670 = patterns (ranked by A333217).
A003242 = anti-run compositions, complement A261983, ranked by A333489.
A005649 = anti-run patterns, complement A069321.
A019536 = necklace patterns.
A025047/A129852/A129853 = alternating compositions, ranked by A345167.
A226316 = patterns avoiding (1,2,3), weakly A052709, complement A335515.
A345163 = normal partitions w/ alternating permutation, complement A345162.
A345170 = partitions w/ alternating permutation, complement A345165.
A349055 = normal multisets w/ alternating permutation, complement A349050.
Cf. A049774, A052955, A096441, A128761, A274230, A335456, A335457, A336103, A344605, A344614.

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

a(n) = A000670(n) - A345194(n).

Terms a(9) and beyond from Andrew Howroyd, Feb 04 2022

A386633 Number of separable type set partitions of {1..n}.

Original entry on oeis.org

1, 1, 1, 4, 10, 46, 166, 827, 3795, 20645, 112124, 672673, 4163743, 27565188, 190168577, 1381763398, 10468226150, 82844940414, 681863474058, 5832378929502, 51720008131148, 474862643822274, 4506628734688128, 44151853623626218, 445956917001833090, 4638586880336637692

Offset: 0

Gus Wiseman, Aug 09 2025

nonn

A set partition is of separable type iff the underlying set has a permutation whose adjacent elements always belong to different blocks. Note that this only depends on the sizes of the blocks.
A set partition is also of separable type iff its greatest block size is at most one more than the sum of all its other block sizes.
This is different from separable partitions (A325534) and partitions of separable type (A336106).

			The a(1) = 1 through a(4) = 10 set partitions:
  {{1}}  {{1},{2}}  {{1},{2,3}}    {{1,2},{3,4}}
                    {{1,2},{3}}    {{1,3},{2,4}}
                    {{1,3},{2}}    {{1,4},{2,3}}
                    {{1},{2},{3}}  {{1},{2},{3,4}}
                                   {{1},{2,3},{4}}
                                   {{1,2},{3},{4}}
                                   {{1},{2,4},{3}}
                                   {{1,3},{2},{4}}
                                   {{1,4},{2},{3}}
                                   {{1},{2},{3},{4}}

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

For separable partitions see A386583, sums A325534, ranks A335433.
For inseparable partitions see A386584, sums A325535, ranks A335448.
For separable type partitions see A386585, sums A336106, ranks A335127.
For inseparable type partitions see A386586, sums A386638 or A025065, ranks A335126.
The complement is counted by A386634, sums of A386636.
Row sums of A386635.
A000110 counts set partitions, row sums of A048993.
A000670 counts ordered set partitions.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A279790 counts disjoint families on strongly normal multisets.
A335434 counts separable factorizations, inseparable A333487.
A336103 counts normal separable multisets, inseparable A336102.
Cf. A001055, A072233, A106351, A190945, A239455, A239964, A335125, A386575, A386578, A386580, A386587.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
stnseps[stn_]:=Select[Permutations[Union@@stn],And@@Table[Position[stn,#[[i]]][[1,1]]!=Position[stn,#[[i+1]]][[1,1]],{i,Length[#]-1}]&]
Table[Length[Select[sps[Range[n]],stnseps[#]!={}&]],{n,0,5}]

a(12)-a(25) from Alois P. Heinz, Aug 10 2025

A035341 Sum of ordered factorizations over all prime signatures with n factors.

Original entry on oeis.org

1, 1, 5, 25, 173, 1297, 12225, 124997, 1492765, 19452389, 284145077, 4500039733, 78159312233, 1460072616929, 29459406350773, 634783708448137, 14613962109584749, 356957383060502945, 9241222160142506097, 252390723655315856437, 7260629936987794508973

Offset: 0

Alford Arnold

Let f(n) = number of ordered factorizations of n (A074206(n)); a(n) = sum of f(k) over all terms k in A025487 that have n factors.
When the unordered spectrum A035310 is so ordered the sequences A000041 A000070 ...A035098 A000110 yield A000079 A001792 ... A005649 A000670 respectively.
Row sums of A095705. - David Wasserman, Feb 22 2008
From Ludovic Schwob, Sep 23 2023: (Start)
a(n) is the number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums. The a(3) = 25 matrices:
  [1 1 1] [1 2] [2 1] [3]
  .
  [1 1] [1 1] [1 1 0] [1 0 1] [0 1 1] [2] [0 2] [2 0]
  [1 0] [0 1] [0 0 1] [0 1 0] [1 0 0] [1] [1 0] [0 1]
  .
  [1] [1 0] [0 1] [1 0] [0 1] [1 0 0] [1 0 0] [0 1] [1 0]
  [1] [1 0] [0 1] [0 1] [1 0] [0 1 0] [0 0 1] [1 0] [0 1]
  [1] [0 1] [1 0] [1 0] [0 1] [0 0 1] [0 1 0] [1 0] [0 1]
  .
  [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [1 0 0] [0 1 0] [1 0 0] (End)

			a(3) = 25 because there are 3 terms in A025487 with 3 factors, namely 8, 12, 30; and f(8)=4, f(12)=8, f(30)=13 and 4+8+13 = 25.

Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 36 terms from David Wasserman)
Eric Weisstein's World of Mathematics, Perfect Partition

Cf. A005651, A025487, A035310, A255906, A365961.

Row sums of A261719.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..n/i)))
    end:
a:= n->add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 29 2015

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k]*If[j == 0, 1, Binomial[i + k - 1, k - 1]^j], {j, 0, n/i}]]];
a[n_] := Sum[Sum[b[n, n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}], {k, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz, updated Dec 15 2020 *)

R(n,k)=Vec(-1 + 1/prod(j=1, n, 1 - binomial(k+j-1,j)*x^j + 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, Sep 23 2023

a(n) ~ c * n! / log(2)^n, where c = 1/(2*log(2)) * Product_{k>=2} 1/(1-1/k!) = A247551 / (2*log(2)) = 1.8246323... . - Vaclav Kotesovec, Jan 21 2017

More terms from Erich Friedman.

More terms from David Wasserman, Feb 22 2008

A349798 Number of weakly alternating ordered prime factorizations of n with at least two adjacent equal parts.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 4, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 5, 1, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 2, 2, 0, 0, 0, 5, 1, 0, 0, 2, 0, 0, 0

Offset: 1

Gus Wiseman, Dec 14 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence counts permutations of prime factors that are weakly but not strongly alternating. Alternating permutations of multisets are a generalization of alternating or up-down permutations of {1..n}.

			Using prime indices instead of factors, the a(n) ordered prime factorizations for selected n are:
n = 4    12    24     48      90     120     192       240      270
   ------------------------------------------------------------------
    11   112   1112   11112   1223   11132   1111112   111132   12232
         211   1121   11121   1322   11213   1111121   111213   13222
               1211   11211   2213   11312   1111211   111312   21223
               2111   12111   2231   21113   1112111   112131   21322
                      21111   3122   21311   1121111   113121   22132
                              3221   23111   1211111   121113   22213
                                     31112   2111111   121311   22231
                                     31211             131112   22312
                                                       131211   23122
                                                       211131   23221
                                                       213111   31222
                                                       231111   32212
                                                       311121
                                                       312111

Wikipedia, Alternating permutation

This is the weakly but not strictly alternating case of A008480.
Including alternating (in fact, anti-run) permutations gives A349056.
These partitions are counted by A349795, ranked by A350137.
A complementary version is A349796, ranked by A350140.
The version for compositions is A349800, ranked by A349799.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A335452 = anti-run ordered prime factorizations.
A344652 = ordered prime factorizations w/o weakly increasing triples.
A345164 = alternating ordered prime factorizations, with twins A344606.
A345194 = alternating patterns, with twins A344605.
A349052/A129852/A129853 = weakly alternating compositions.
A349053 = non-weakly alternating compositions, ranked by A349057.
A349060 = weakly alternating partitions, complement A349061.
A349797 = non-weakly alternating ordered prime factorizations.
Cf. A005649, A096441, A335433, A335448, A344614, A344615, A345166, A345173, A345195, A348609, A349794, A349801.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Permutations[primeMS[n]],(whkQ[#]||whkQ[-#])&&MatchQ[#,{_,x_,x_,_}]&]],{n,100}]

A349799 Numbers k such that the k-th composition in standard order is weakly alternating but has at least two adjacent equal parts.

Original entry on oeis.org

3, 7, 10, 11, 14, 15, 19, 21, 23, 26, 27, 28, 29, 30, 31, 35, 36, 39, 42, 43, 47, 51, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 73, 74, 79, 83, 84, 85, 86, 87, 90, 91, 94, 95, 99, 100, 103, 106, 111, 112, 113, 114, 115, 118, 119, 120, 121, 122, 123, 124, 125

Offset: 1

Gus Wiseman, Dec 15 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.
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.
This sequence ranks compositions that are weakly but not strongly alternating.

			The terms and corresponding compositions begin:
   3: (1,1)
   7: (1,1,1)
  10: (2,2)
  11: (2,1,1)
  14: (1,1,2)
  15: (1,1,1,1)
  19: (3,1,1)
  21: (2,2,1)
  23: (2,1,1,1)
  26: (1,2,2)
  27: (1,2,1,1)
  28: (1,1,3)
  29: (1,1,2,1)
  30: (1,1,1,2)
  31: (1,1,1,1,1)

Wikipedia, Alternating permutation

Partitions of this type are counted by A349795, ranked by A350137.
Permutations of prime indices of this type are counted by A349798.
These compositions are counted by A349800.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A003242 = Carlitz (anti-run) compositions, ranked by A333489.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A261983 = non-anti-run compositions, ranked by A348612.
A345164 = alternating permutations of prime indices, with twins A344606.
A345165 = partitions without an alternating permutation, ranked by A345171.
A345170 = partitions with an alternating permutation, ranked by A345172.
A345166 = separable partitions with no alternations, ranked by A345173.
A345192 = non-alternating compositions, ranked by A345168.
A345195 = non-alternating anti-run compositions, ranked by A345169.
A349052/A129852/A129853 = weakly alternating compositions.
A349053 = non-weakly alternating compositions, ranked by A349057.
A349056 = weak alternations of prime indices, complement A349797.
A349060 = weak alternations of partitions, complement A349061.
Cf. A005649, A049774, A096441, A128761, A344615, A345163, A349054, A349058, A349796, A349801, A350140.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Select[Range[0,100],(whkQ[stc[#]]||whkQ[-stc[#]])&&MatchQ[stc[#],{_,x_,x_,_}]&]

Equals A345168 \ A349057 = A348612 \ A349057.

A386635 Triangle read by rows where T(n,k) is the number of separable type set partitions of {1..n} into k blocks.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 3, 1, 0, 0, 3, 6, 1, 0, 0, 10, 25, 10, 1, 0, 0, 10, 75, 65, 15, 1, 0, 0, 35, 280, 350, 140, 21, 1, 0, 0, 35, 770, 1645, 1050, 266, 28, 1, 0, 0, 126, 2737, 7686, 6951, 2646, 462, 36, 1, 0, 0, 126, 7455, 32725, 42315, 22827, 5880, 750, 45, 1

Offset: 0

Gus Wiseman, Aug 10 2025

A set partition is of separable type iff the underlying set has a permutation whose adjacent elements always belong to different blocks. Note that this only depends on the sizes of the blocks.
A set partition is also of separable type iff its greatest block size is at most one more than the sum of all its other blocks sizes.
This is different from separable partitions (A325534) and partitions of separable type (A336106).

			Row n = 4 counts the following set partitions:
  .  .  {{1,2},{3,4}}  {{1},{2},{3,4}}  {{1},{2},{3},{4}}
        {{1,3},{2,4}}  {{1},{2,3},{4}}
        {{1,4},{2,3}}  {{1},{2,4},{3}}
                       {{1,2},{3},{4}}
                       {{1,3},{2},{4}}
                       {{1,4},{2},{3}}
Triangle begins:
    1
    0    1
    0    0    1
    0    0    3    1
    0    0    3    6    1
    0    0   10   25   10    1
    0    0   10   75   65   15    1
    0    0   35  280  350  140   21    1

Column k = 2 appears to be A128015.
For separable partitions we have A386583, sums A325534, ranks A335433.
For inseparable partitions we have A386584, sums A325535, ranks A335448.
For separable type partitions we have A386585, sums A336106, ranks A335127.
For inseparable type partitions we have A386586, sums A386638 or A025065, ranks A335126.
Row sums are A386633.
The complement is counted by A386636, row sums A386634.
A000110 counts set partitions, row sums of A048993.
A000670 counts ordered set partitions.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A335434 counts separable factorizations, inseparable A333487.
A336103 counts normal separable multisets, inseparable A336102.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
A386587 counts disjoint families of strict partitions of each prime exponent.
Cf. A072233, A097805, A106351, A116861, A190945, A279790, A333382, A335125, A374252, A386575, A386578.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
stnseps[stn_]:=Select[Permutations[Union@@stn],And@@Table[Position[stn,#[[i]]][[1,1]]!=Position[stn,#[[i+1]]][[1,1]],{i,Length[#]-1}]&];
Table[Length[Select[sps[Range[n]],Length[#]==k&&stnseps[#]!={}&]],{n,0,5},{k,0,n}]

A386636 Triangle read by rows where T(n,k) is the number of inseparable type set partitions of {1..n} into k blocks.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4, 0, 0, 0, 1, 5, 0, 0, 0, 0, 1, 21, 15, 0, 0, 0, 0, 1, 28, 21, 0, 0, 0, 0, 0, 1, 92, 196, 56, 0, 0, 0, 0, 0, 1, 129, 288, 84, 0, 0, 0, 0, 0, 0, 1, 385, 1875, 1380, 210, 0, 0, 0, 0, 0, 0, 1, 561, 2860, 2145, 330, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Aug 10 2025

A set partition is of inseparable type iff the underlying set has no permutation whose adjacent elements always belong to different blocks. Note that this only depends on the sizes of the blocks.
A set partition is also of inseparable type iff its greatest block size is at least 2 more than the sum of all its other block sizes.
This is different from inseparable partitions (A325535) and partitions of inseparable type (A386638 or A025065).

			Row n = 6 counts the following set partitions:
  .  {123456}  {1}{23456}  {1}{2}{3456}  .  .  .
               {12}{3456}  {1}{2345}{6}
               {13}{2456}  {1}{2346}{5}
               {14}{2356}  {1}{2356}{4}
               {15}{2346}  {1}{2456}{3}
               {16}{2345}  {1234}{5}{6}
               {1234}{56}  {1235}{4}{6}
               {1235}{46}  {1236}{4}{5}
               {1236}{45}  {1245}{3}{6}
               {1245}{36}  {1246}{3}{5}
               {1246}{35}  {1256}{3}{4}
               {1256}{34}  {1345}{2}{6}
               {1345}{26}  {1346}{2}{5}
               {1346}{25}  {1356}{2}{4}
               {1356}{24}  {1456}{2}{3}
               {1456}{23}
               {12345}{6}
               {12346}{5}
               {12356}{4}
               {12456}{3}
               {13456}{2}
Triangle begins:
    0
    0    0
    0    1    0
    0    1    0    0
    0    1    4    0    0
    0    1    5    0    0    0
    0    1   21   15    0    0    0
    0    1   28   21    0    0    0    0
    0    1   92  196   56    0    0    0    0
    0    1  129  288   84    0    0    0    0    0
    0    1  385 1875 1380  210    0    0    0    0    0

For separable partitions we have A386583, sums A325534, ranks A335433.
For inseparable partitions we have A386584, sums A325535, ranks A335448.
For separable type partitions we have A386585, sums A336106, ranks A335127.
For inseparable type partitions we have A386586, sums A386638 or A025065, ranks A335126.
Row sums are A386634.
The complement is counted by A386635, row sums A386633.
A000110 counts set partitions, row sums of A048993.
A000670 counts ordered set partitions.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A279790 counts disjoint families on strongly normal multisets.
A335434 counts separable factorizations, inseparable A333487.
A336103 counts normal separable multisets, inseparable A336102.
A386587 counts disjoint families of strict partitions of each prime exponent.
Cf. A008284, A072233, A097805, A106351, A116861, A124762, A129506, A190945, A239455, A335125, A374252, A386575.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
stnseps[stn_]:=Select[Permutations[Union@@stn],And@@Table[Position[stn,#[[i]]][[1,1]]!=Position[stn,#[[i+1]]][[1,1]],{i,Length[#]-1}]&]
Table[Length[Select[sps[Range[n]],Length[#]==k&&stnseps[#]=={}&]],{n,0,5},{k,0,n}]

cp's OEIS Frontend

A226515 Row 2 of array in A226513.

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A386634 Number of inseparable type set partitions of {1..n}.

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A335407 Number of anti-run permutations of the prime indices of n!.

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A350252 Number of non-alternating patterns of length n.

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A386633 Number of separable type set partitions of {1..n}.

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A035341 Sum of ordered factorizations over all prime signatures with n factors.

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A349798 Number of weakly alternating ordered prime factorizations of n with at least two adjacent equal parts.

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