A268193 -id:A268193 - cp's OEIS frontend

A384175 Number of subsets of {1..n} with all distinct lengths of maximal runs (increasing by 1).

1, 2, 4, 7, 13, 24, 44, 77, 135, 236, 412, 713, 1215, 2048, 3434, 5739, 9559, 15850, 26086, 42605, 69133, 111634, 179602, 288069, 460553, 733370, 1162356, 1833371, 2878621, 4501856, 7016844, 10905449, 16904399, 26132460, 40279108, 61885621, 94766071, 144637928

Offset: 0

Gus Wiseman, Jun 16 2025

nonn

			The subset {2,3,5,6,7,9} has maximal runs ((2,3),(5,6,7),(9)), with lengths (2,3,1), so is counted under a(9).
The a(0) = 1 through a(4) = 13 subsets:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {2}    {2}      {2}
           {1,2}  {3}      {3}
                  {1,2}    {4}
                  {2,3}    {1,2}
                  {1,2,3}  {2,3}
                           {3,4}
                           {1,2,3}
                           {1,2,4}
                           {1,3,4}
                           {2,3,4}
                           {1,2,3,4}

Christian Sievers, Table of n, a(n) for n = 0..1000

For equal instead of distinct lengths we have A243815.
These subsets are ranked by A328592.
The complement is counted by A384176.
For anti-runs instead of runs we have A384177, ranks A384879.
For partitions instead of subsets we have A384884, A384178, A384886, A384880.
For permutations instead of subsets we have A384891, equal instead of distinct A384892.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000009, A010027, A044813, A047993, A242882, A325325, A329739, A351202, A383013, A384889, A384890.

Table[Length[Select[Subsets[Range[n]],UnsameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]

lista(n)={my(o=(1-x^(n+1))/(1-x)*O(y^(n+2)),p=prod(i=1,n,1+o+x*y^(i+1)/(1-y),1/(1-y)));p=subst(serlaplace(p),x,1);Vec(p-1)} \\ Christian Sievers, Jun 18 2025

a(21) and beyond from Christian Sievers, Jun 18 2025

A384893 Triangle read by rows where T(n,k) is the number of subsets of {1..n} with k maximal anti-runs (increasing by more than 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 7, 5, 2, 1, 1, 12, 10, 6, 2, 1, 1, 20, 20, 13, 7, 2, 1, 1, 33, 38, 29, 16, 8, 2, 1, 1, 54, 71, 60, 39, 19, 9, 2, 1, 1, 88, 130, 122, 86, 50, 22, 10, 2, 1, 1, 143, 235, 241, 187, 116, 62, 25, 11, 2, 1, 1, 232, 420, 468, 392, 267, 150, 75, 28, 12, 2, 1

Offset: 0

Gus Wiseman, Jun 21 2025

			The subset {3,6,7,9,11,12} has maximal anti-runs ((3,6),(7,9,11),(12)), so is counted under T(12,3).
The subset {3,6,7,9,10,12} has maximal anti-runs ((3,6),(7,9),(10,12)), so is counted under T(12,3).
Row n = 5 counts the following subsets:
  {}  {1}      {1,2}    {1,2,3}    {1,2,3,4}  {1,2,3,4,5}
      {2}      {2,3}    {2,3,4}    {2,3,4,5}
      {3}      {3,4}    {3,4,5}
      {4}      {4,5}    {1,2,3,5}
      {5}      {1,2,4}  {1,2,4,5}
      {1,3}    {1,2,5}  {1,3,4,5}
      {1,4}    {1,3,4}
      {1,5}    {1,4,5}
      {2,4}    {2,3,5}
      {2,5}    {2,4,5}
      {3,5}
      {1,3,5}
Triangle begins:
   1
   1   1
   1   2   1
   1   4   2   1
   1   7   5   2   1
   1  12  10   6   2   1
   1  20  20  13   7   2   1
   1  33  38  29  16   8   2   1
   1  54  71  60  39  19   9   2   1
   1  88 130 122  86  50  22  10   2   1
   1 143 235 241 187 116  62  25  11   2   1
   1 232 420 468 392 267 150  75  28  12   2   1
   1 376 744 894 806 588 363 188  89  31  13   2   1

Column k = 1 is A000071.
Row sums are A000079.
Column k = 2 is A001629.
For runs instead of anti-runs we have A034839, for strict partitions A116674.
The case containing n is A053538.
For integer partitions instead of subsets we have A268193, strict A384905.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
Cf. A010027, A384177, A384879, A384889, A384890.

Table[Length[Select[Subsets[Range[n]],Length[Split[#,#2!=#1+1&]]==k&]],{n,0,10},{k,0,n}]

A116674 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts and having exactly k distinct parts (n>=1, k>=1).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 3, 1, 5, 3, 4, 1, 2, 7, 1, 2, 8, 2, 2, 10, 3, 2, 11, 5, 2, 13, 7, 4, 12, 11, 1, 19, 11, 1, 2, 18, 17, 1, 3, 20, 21, 2, 2, 22, 27, 3, 2, 25, 32, 5, 4, 24, 41, 7, 2, 30, 46, 11, 2, 31, 56, 15, 2, 36, 62, 22, 3, 33, 80, 25, 1, 2, 39, 87, 36, 1, 4, 38, 103, 45, 2, 2, 45

Offset: 1

Emeric Deutsch, Feb 22 2006

Row n has floor(sqrt(n)) terms. Row sums yield A000009. T(n,1)=A001227(n) (n>=1). Sum(k*T(n,k),k>=1)=A038348(n-1) (n>=1).

Conjecture: Also the number of strict integer partitions of n with k maximal runs of consecutive parts decreasing by 1. - Gus Wiseman, Jun 24 2025

			From _Gus Wiseman_, Jun 24 2025: (Start)
Triangle begins:
   1:  1
   2:  1
   3:  2
   4:  1  1
   5:  2  1
   6:  2  2
   7:  2  3
   8:  1  5
   9:  3  4  1
  10:  2  7  1
  11:  2  8  2
  12:  2 10  3
  13:  2 11  5
  14:  2 13  7
  15:  4 12 11
  16:  1 19 11  1
  17:  2 18 17  1
  18:  3 20 21  2
  19:  2 22 27  3
  20:  2 25 32  5
Row n = 9 counts the following partitions into odd parts by number of distinct parts:
  (9)                  (7,1,1)          (5,3,1)
  (3,3,3)              (3,3,1,1,1)
  (1,1,1,1,1,1,1,1,1)  (5,1,1,1,1)
                       (3,1,1,1,1,1,1)
Row n = 9 counts the following strict partitions by number of maximal runs:
  (9)      (6,3)    (5,3,1)
  (5,4)    (7,2)
  (4,3,2)  (8,1)
           (6,2,1)
(End)

Alois P. Heinz, Rows n = 1..1000, flattened

Row sums are A000009, strict case of A000041.
Row lengths are A000196.
Leading terms are A001227.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat partitions, ranks A066311 or A073491.
A047993 counts partitions with max part = length.
A152140 counts partitions into odd parts by length.
A268193 counts partitions by number of maximal anti-runs, strict A384905.
A384881 counts partitions by number of maximal runs.
Cf. A008284, A038348, A047966, A089259, A325325, A384178, A384886, A384887.

g:=product(1+t*x^(2*j-1)/(1-x^(2*j-1)),j=1..35): gser:=simplify(series(g,x=0,34)): for n from 1 to 29 do P[n]:=coeff(gser,x^n) od: for n from 1 to 29 do seq(coeff(P[n],t,j),j=1..floor(sqrt(n))) od; # yields sequence in triangular form
# second Maple program:
with(numtheory):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, i-2)*`if`(j=0, 1, x), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(
         b(n, iquo(n+1, 2)*2-1)):
seq(T(n), n=1..30);  # Alois P. Heinz, Mar 08 2015

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-2]*If[j == 0, 1, x], {j, 0, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, Quotient[n+1, 2]*2-1]]; Table[T[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],OddQ[Times@@#]&&Length[Union[#]]==k&]],{n,1,12},{k,1,Floor[Sqrt[n]]}] (*  Gus Wiseman, Jun 24 2025 *)

G.f.: product(1+tx^(2j-1)/(1-x^(2j-1)), j=1..infinity).

A384177 Number of subsets of {1..n} with all distinct lengths of maximal anti-runs (increasing by more than 1).

Original entry on oeis.org

1, 2, 3, 5, 10, 19, 35, 62, 109, 197, 364, 677, 1251, 2288, 4143, 7443, 13318, 23837, 42809, 77216, 139751, 253293, 458800, 829237, 1494169, 2683316, 4804083, 8580293, 15301324, 27270061, 48607667, 86696300, 154758265, 276453311, 494050894, 882923051

Offset: 0

Gus Wiseman, Jun 16 2025

nonn

			The subset {1,2,4,5,7,10} has maximal anti-runs ((1),(2,4),(5,7,10)), with lengths (1,2,3), so is counted under a(10).
The a(0) = 1 through a(5) = 19 subsets:
  {}  {}   {}   {}     {}       {}
      {1}  {1}  {1}    {1}      {1}
           {2}  {2}    {2}      {2}
                {3}    {3}      {3}
                {1,3}  {4}      {4}
                       {1,3}    {5}
                       {1,4}    {1,3}
                       {2,4}    {1,4}
                       {1,2,4}  {1,5}
                       {1,3,4}  {2,4}
                                {2,5}
                                {3,5}
                                {1,2,4}
                                {1,2,5}
                                {1,3,4}
                                {1,3,5}
                                {1,4,5}
                                {2,3,5}
                                {2,4,5}

Christian Sievers, Table of n, a(n) for n = 0..1000

For runs instead of anti-runs we have A384175, complement A384176.
These subsets are ranked by A384879.
For strict partitions instead of subsets we have A384880, see A384178, A384884, A384886.
For equal instead of distinct lengths we have A384889, for runs A243815.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000009, A010027, A044813, A047993, A106529, A123513, A242882, A325325, A328592, A329739, A351202, A384890.

Table[Length[Select[Subsets[Range[n]],UnsameQ@@Length/@Split[#,#2!=#1+1&]&]],{n,0,10}]

lista(n)={my(o=(1-x^(n+1))/(1-x)*O(y*y^n),p=prod(i=1,(n+1)\2,1+o+x*y^(2*i-1)/(1-y)^(i-1)));p=subst(serlaplace(p),x,1);Vec((p-y)/(1-y)^2)} \\ Christian Sievers, Jun 18 2025

a(21) and beyond from Christian Sievers, Jun 18 2025

A384176 Number of subsets of {1..n} without all distinct lengths of maximal runs (increasing by 1).

Original entry on oeis.org

0, 0, 0, 1, 3, 8, 20, 51, 121, 276, 612, 1335, 2881, 6144, 12950, 27029, 55977, 115222, 236058, 481683, 979443

Offset: 0

Gus Wiseman, Jun 16 2025

			The subset {1,3,4,8,9} has maximal runs ((1),(3,4),(8,9)), with lengths (1,2,2), so is counted under a(10).
The a(0) = 0 through a(6) = 20 subsets:
  .  .  .  {1,3}  {1,3}  {1,3}      {1,3}
                  {1,4}  {1,4}      {1,4}
                  {2,4}  {1,5}      {1,5}
                         {2,4}      {1,6}
                         {2,5}      {2,4}
                         {3,5}      {2,5}
                         {1,3,5}    {2,6}
                         {1,2,4,5}  {3,5}
                                    {3,6}
                                    {4,6}
                                    {1,3,5}
                                    {1,3,6}
                                    {1,4,6}
                                    {2,4,6}
                                    {1,2,4,5}
                                    {1,2,4,6}
                                    {1,2,5,6}
                                    {1,3,4,6}
                                    {1,3,5,6}
                                    {2,3,5,6}

For equal instead of distinct lengths the complement is A243815.
These subsets are ranked by the non-members of A328592.
The complement is counted by A384175.
For strict partitions instead of subsets see A384178, A384884, A384886, A384880.
For permutations instead of subsets see A384891, A384892, A010027.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000009, A044813, A242882, A325325, A329739, A351202, A383013, A384177, A384889, A384890.

Table[Length[Select[Subsets[Range[n]],!UnsameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]

A384905 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with k maximal anti-runs (decreasing by more than 1).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 5, 2, 1, 0, 0, 0, 0, 0, 0, 0, 6, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 7, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Jun 21 2025

			The T(10,2) = 3 strict partitions with 2 maximal anti-runs are: (7,2,1), (5,4,1), (5,3,2).
Triangle begins:
  1
  0  1
  0  1  0
  0  1  1  0
  0  2  0  0  0
  0  2  1  0  0  0
  0  3  0  1  0  0  0
  0  3  2  0  0  0  0  0
  0  4  2  0  0  0  0  0  0
  0  5  2  1  0  0  0  0  0  0
  0  6  3  0  1  0  0  0  0  0  0
  0  7  4  1  0  0  0  0  0  0  0  0
  0  9  3  3  0  0  0  0  0  0  0  0  0

Row sums are A000009.
Column k = 1 is A003114.
For subsets instead of strict integer partitions see A053538, A119900, A210034.
For runs instead of anti-runs we have A116674, for subsets A034839.
This is the strict case of A268193.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
Cf. A010027, A384177, A384886, A384889, A384890.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[Split[#,#1!=#2+1&]]==k&]],{n,0,10},{k,0,n}]

A116931 Number of partitions of n in which each part, with the possible exception of the largest, occurs at least twice.

Original entry on oeis.org

1, 2, 2, 4, 4, 8, 8, 13, 15, 22, 24, 37, 40, 57, 64, 89, 98, 135, 149, 199, 224, 292, 325, 424, 472, 601, 676, 850, 950, 1191, 1329, 1643, 1845, 2258, 2524, 3082, 3442, 4158, 4659, 5591, 6246, 7472, 8338, 9903, 11072, 13077, 14586, 17184, 19150, 22431, 25019

Offset: 1

Emeric Deutsch, Feb 27 2006

nonn

Also, partitions of n in which any two distinct parts differ by at least 2. Example: a(5) = 4 because we have [5], [4,1], [3,1,1] and [1,1,1,1,1].

			a(5) = 4 because we have [5], [3,1,1], [2,1,1,1] and [1,1,1,1,1].
q + 2*q^2 + 2*q^3 + 4*q^4 + 4*q^5 + 8*q^6 + 8*q^7 + 13*q^8 + 15*q^9 + ...
There are a(9) = 15 partitions of 9 where distinct parts differ by at least 2:
01:  [ 1 1 1 1 1 1 1 1 1 ]
02:  [ 3 1 1 1 1 1 1 ]
03:  [ 3 3 1 1 1 ]
04:  [ 3 3 3 ]
05:  [ 4 1 1 1 1 1 ]
06:  [ 4 4 1 ]
07:  [ 5 1 1 1 1 ]
08:  [ 5 2 2 ]
09:  [ 5 3 1 ]
10:  [ 6 1 1 1 ]
11:  [ 6 3 ]
12:  [ 7 1 1 ]
13:  [ 7 2 ]
14:  [ 8 1 ]
15:  [ 9 ]
- _Joerg Arndt_, Jun 09 2013

P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 52, Article 298.

Vaclav Kotesovec, Table of n, a(n) for n = 1..20000 (terms 1..7500 from Alois P. Heinz)
Mingjia Yang, Doron Zeilberger, Systematic Counting of Restricted Partitions, arXiv:1910.08989 [math.CO], 2019.

Cf. A116932, A007690.
Column k=2 of A218698. - Alois P. Heinz, Nov 04 2012
Column k=0 of A268193. - Alois P. Heinz, Feb 13 2016

g:=sum(x^k*product(1+x^(2*j)/(1-x^j),j=1..k-1)/(1-x^k),k=1..70): gser:=series(g,x=0,60): seq(coeff(gser,x^n),n=1..54);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-2), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=1..70);  # Alois P. Heinz, Nov 04 2012

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-2], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *)

{a(n) = if( n<1, 0, polcoeff( sum( k=1, n, x^k / (1 - x^k) * prod( j=1, k-1, 1 + x^(2*j) / (1 - x^j), 1 + x * O(x^(n-k)))), n))} /* Michael Somos, Jan 26 2008 */

G.f.: sum(x^k*product(1+x^(2j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity). More generally, the g.f. of partitions of n in which each part, with the possible exception of the largest, occurs at least b times, is sum(x^k*product(1+x^(bj)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity). It is also the g.f. of partitions of n in which any two distinct parts differ by at least b.

log(a(n)) ~ 2*Pi*sqrt(n)/3. - Vaclav Kotesovec, Jan 28 2022

A243815 Number of length n words on alphabet {0,1} such that the length of every maximal block of 0's (runs) is the same.

Original entry on oeis.org

1, 2, 4, 8, 14, 24, 39, 62, 97, 151, 233, 360, 557, 864, 1344, 2099, 3290, 5176, 8169, 12931, 20524, 32654, 52060, 83149, 133012, 213069, 341718, 548614, 881572, 1417722, 2281517, 3673830, 5918958, 9540577, 15384490, 24817031, 40045768, 64637963, 104358789

Offset: 0

Geoffrey Critzer, Jun 11 2014

nonn

Number of terms of A164710 with exactly n+1 binary digits. - Robert Israel, Nov 09 2015
From Gus Wiseman, Jun 23 2025: (Start)
This is the number of subsets of {1..n} with all equal lengths of runs of consecutive elements increasing by 1. For example, the runs of S = {1,2,5,6,8,9} are ((1,2),(5,6),(8,9)), with lengths (2,2,2), so S is counted under a(9). The a(0) = 1 through a(4) = 14 subsets are:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {2}    {2}      {2}
           {1,2}  {3}      {3}
                  {1,2}    {4}
                  {1,3}    {1,2}
                  {2,3}    {1,3}
                  {1,2,3}  {1,4}
                           {2,3}
                           {2,4}
                           {3,4}
                           {1,2,3}
                           {2,3,4}
                           {1,2,3,4}
(End)

			0110 is a "good" word because the length of both its runs of 0's is 1.
Words of the form 11...1 are good words because the condition is vacuously satisfied.
a(5) = 24 because there are 32 length 5 binary words but we do not count: 00010, 00101, 00110, 01000, 01001, 01100, 10010, 10100.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A164710.
These subsets are ranked by A164707, complement A164708.
For distinct instead of equal lengths we have A384175, complement A384176.
For anti-runs instead of runs we have A384889, for partitions A384888.
For permutations instead of subsets we have A384892, distinct instead of equal A384891.
For partitions instead of subsets we have A384904, strict A384886.
The complement is counted by A385214.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A049988 counts partitions with equal run-lengths, distinct A325325.
A329738 counts compositions with equal run-lengths, distinct A329739.
A384887 counts partitions with equal lengths of gapless runs, distinct A384884.
Cf. A010027, A044813, A069010, A268193, A383013, A384177, A384893.

a:= n-> 1 + add(add((d-> binomial(d+j, d))(n-(i*j-1))
          , j=1..iquo(n+1, i)), i=2..n+1):
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 11 2014

nn=30;Prepend[Map[Total,Transpose[Table[Drop[CoefficientList[Series[ (1+x^k)/(1-x-x^(k+1))-1/(1-x),{x,0,nn}],x],1],{k,1,nn}]]],0]+1
Table[Length[Select[Subsets[Range[n]],SameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}] (* Gus Wiseman, Jun 23 2025 *)

A053538 Triangle: a(n,m) = ways to place p balls in n slots with m in the rightmost p slots, 0<=p<=n, 0<=m<=n, summed over p, a(n,m)= Sum_{k=0..n} binomial(k,m)*binomial(n-k,k-m), (see program line).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 4, 1, 1, 8, 10, 7, 5, 1, 1, 13, 18, 16, 9, 6, 1, 1, 21, 33, 31, 23, 11, 7, 1, 1, 34, 59, 62, 47, 31, 13, 8, 1, 1, 55, 105, 119, 101, 66, 40, 15, 9, 1, 1, 89, 185, 227, 205, 151, 88, 50, 17, 10, 1, 1, 144, 324, 426, 414, 321, 213, 113, 61, 19, 11, 1, 1

Offset: 0

Wouter Meeussen, May 23 2001

Riordan array (1/(1-x-x^2), x(1-x)/(1-x-x^2)). Row sums are A000079. Diagonal sums are A006053(n+2). - Paul Barry, Nov 01 2006
Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 05 2012
Mirror image of triangle in A208342. - Philippe Deléham, Mar 05 2012
A053538 is jointly generated with A076791 as an array of coefficients of polynomials u(n,x):  initially, u(1,x)=v(1,x)=1, for n>1, u(n,x) = x*u(n-1,x) + v(n-1,x) and v(n,x) = u(n-1,x) + v(n-1,x).  See the Mathematica section at A076791. - Clark Kimberling, Mar 08 2012
The matrix inverse starts
    1;
   -1,   1;
   -1,  -1,   1;
    1,  -2,  -1,  1;
    3,   1,  -3, -1,  1;
    1,   6,   1, -4, -1,  1;
   -7,   4,  10,  1, -5, -1,  1;
  -13, -13,   8, 15,  1, -6, -1,  1;
    3, -31, -23, 13, 21,  1, -7, -1, 1; - R. J. Mathar, Mar 15 2013
Also appears to be the number of subsets of {1..n} containing n with k maximal anti-runs of consecutive elements increasing by more than 1. For example, the subset {1,3,6,7,11,12} has maximal anti-runs ((1,3,6),(7,11),(12)) so is counted under a(12,3). For runs instead of anti-runs we get A202064. - Gus Wiseman, Jun 26 2025

			n=4; Table[binomial[k, j]binomial[n-k, k-j], {k, 0, n}, {j, 0, n}] splits {1, 4, 6, 4, 1} into {{1, 0, 0, 0, 0}, {3, 1, 0, 0, 0}, {1, 4, 1, 0, 0}, {0, 0, 3, 1, 0}, {0, 0, 0, 0, 1}} and this gives summed by columns {5, 5, 4, 1, 1}
Triangle begins :
   1;
   1,  1;
   2,  1,  1;
   3,  3,  1, 1;
   5,  5,  4, 1, 1;
   8, 10,  7, 5, 1, 1;
  13, 18, 16, 9, 6, 1, 1;
...
(0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, ...) begins :
  1;
  0,  1;
  0,  1,  1;
  0,  2,  1,  1;
  0,  3,  3,  1, 1;
  0,  5,  5,  4, 1, 1;
  0,  8, 10,  7, 5, 1, 1;
  0, 13, 18, 16, 9, 6, 1, 1;

Alois P. Heinz, Rows n = 0..140, flattened
R. P. Grimaldi, Extraordinary subsets: a generalization, Fib. Quart., 55 (No. 3, 2017), 114-122. See Table 1.

Column k = 1 is A000045.
Row sums are A000079.
Column k = 2 is A010049.
For runs instead of anti-runs we have A202064.
For integer partitions see A268193, strict A384905, runs A116674.
A034839 counts subsets by number of maximal runs.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
A384893 counts subsets by number of maximal anti-runs.
Cf. A000071, A001629, A010027, A208342, A384177, A384879, A384889, A384890.

Flat(List([0..12], n-> List([0..n], k-> Sum([0..n], j->  Binomial(j,k)*Binomial(n-j,j-k)) ))); # G. C. Greubel, May 16 2019

[[(&+[Binomial(j,k)*Binomial(n-j,j-k): j in [0..n]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 16 2019

a:= (n, m)-> add(binomial(k, m)*binomial(n-k, k-m), k=0..n):
seq(seq(a(n,m), m=0..n), n=0..12);  # Alois P. Heinz, Sep 19 2013

Table[Sum[Binomial[k, m]*Binomial[n-k, k-m], {k,0,n}], {n,0,12}, {m,0,n}]

{T(n,k) = sum(j=0,n, binomial(j,k)*binomial(n-j,j-k))}; \\ G. C. Greubel, May 16 2019

[[sum(binomial(j,k)*binomial(n-j,j-k) for j in (0..n)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 16 2019

From Philippe Deléham, Mar 05 2012: (Start)
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if k>n.
G.f.: 1/(1-(1+y)*x-(1-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A077957(n), A000045(n+1), A000079(n), A001906(n+1), A007070(n), A116415(n), A084326(n+1), A190974(n+1), A190978(n+1), A190984(n+1), A190990(n+1), A190872(n+1) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively. (End)

A384889 Number of subsets of {1..n} with all equal lengths of maximal anti-runs (increasing by more than 1).

Original entry on oeis.org

1, 2, 4, 8, 14, 23, 37, 59, 93, 146, 230, 365, 584, 940, 1517, 2450, 3959, 6404, 10373, 16822, 27298, 44297, 71843, 116429, 188550, 305200, 493930, 799422, 1294108, 2095291, 3392736, 5493168, 8892148, 14390372, 23282110, 37660759, 60914308, 98528312, 159386110

Offset: 0

Gus Wiseman, Jun 18 2025

nonn

			The subset {3,6,7,9,10,12} has maximal anti-runs ((3,6),(7,9),(10,12)), with lengths (2,2,2), so is counted under a(12).
The a(0) = 1 through a(4) = 14 subsets:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {2}    {2}      {2}
           {1,2}  {3}      {3}
                  {1,2}    {4}
                  {1,3}    {1,2}
                  {2,3}    {1,3}
                  {1,2,3}  {1,4}
                           {2,3}
                           {2,4}
                           {3,4}
                           {1,2,3}
                           {2,3,4}
                           {1,2,3,4}

Christian Sievers, Table of n, a(n) for n = 0..1000

For runs instead of anti-runs we have A243815, distinct A384175, complement A384176.
For distinct instead or equal lengths we have A384177, ranks A384879.
For partitions instead of subsets we have A384888.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A047966 counts uniform partitions (equal multiplicities), ranks A072774.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A010027, A044813, A047993, A356606, A384880, A384886, A384890.

Table[Length[Select[Subsets[Range[n]],SameQ@@Length/@Split[#,#2!=#1+1&]&]],{n,0,10}]

lista(n)=Vec(sum(i=1,(n+1)\2,1/(1-x^(2*i-1)/(1-x)^(i-1))-1,1-x+O(x*x^n))/(1-x)^2) \\ Christian Sievers, Jun 20 2025

G.f.: ( Sum_{i>=1} (1/(1-x^(2*i-1)/(1-x)^(i-1))-1) + 1-x ) / (1-x)^2. - Christian Sievers, Jun 21 2025

a(21) and beyond from Christian Sievers, Jun 20 2025

cp's OEIS Frontend

A384175 Number of subsets of {1..n} with all distinct lengths of maximal runs (increasing by 1).

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A384893 Triangle read by rows where T(n,k) is the number of subsets of {1..n} with k maximal anti-runs (increasing by more than 1).

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A116674 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts and having exactly k distinct parts (n>=1, k>=1).

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A384177 Number of subsets of {1..n} with all distinct lengths of maximal anti-runs (increasing by more than 1).

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A384176 Number of subsets of {1..n} without all distinct lengths of maximal runs (increasing by 1).

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A384905 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with k maximal anti-runs (decreasing by more than 1).

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A116931 Number of partitions of n in which each part, with the possible exception of the largest, occurs at least twice.

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A243815 Number of length n words on alphabet {0,1} such that the length of every maximal block of 0's (runs) is the same.

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