cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A268193 Triangle read by rows: T(n,k) (n>=1, k>=0) is the number of partitions of n which have k distinct parts i such that i+1 is also a part.

Original entry on oeis.org

1, 2, 2, 1, 4, 1, 4, 3, 8, 2, 1, 8, 6, 1, 13, 7, 2, 15, 11, 4, 22, 15, 4, 1, 24, 24, 7, 1, 37, 26, 12, 2, 40, 42, 16, 3, 57, 50, 22, 6, 64, 72, 33, 6, 1, 89, 84, 46, 11, 1, 98, 122, 60, 15, 2, 135, 141, 82, 24, 3, 149, 198, 106, 32, 5, 199, 231, 144, 45, 8, 224, 309, 187, 61, 10, 1
Offset: 1

Views

Author

Emeric Deutsch, Feb 13 2016

Keywords

Comments

T(n,k) = number of partitions of n having k singleton parts other than the largest part. Example: T(5,1) = 3 because we have [4,1'], [3,2'], [2,2,1'] (the counted singletons are marked). These partitions are connected by conjugation to those in the definition.
From Gus Wiseman, Jul 10 2025: (Start)
Also the number of integer partitions of n with k maximal subsequences of consecutive parts not decreasing by 1 (anti-runs). For example, row n = 8 counts partitions with the following anti-runs:
((8)) ((3,3),(2)) ((3),(2,2),(1))
((4,4)) ((4),(3,1)) ((3),(2),(1,1,1))
((5,3)) ((5,2),(1))
((6,2)) ((4,2),(1,1))
((7,1)) ((2,2,2),(1,1))
((4,2,2)) ((2,2),(1,1,1,1))
((6,1,1)) ((2),(1,1,1,1,1,1))
((2,2,2,2))
((3,3,1,1))
((5,1,1,1))
((4,1,1,1,1))
((3,1,1,1,1,1))
((1,1,1,1,1,1,1,1))
(End)

Examples

			T(5,1) = 3 because we have [3,2], [2,2,1], and [2,1,1,1].
T(9,2) = 4 because we have [3,2',1,1,1,1'], [3,2,2',1,1'], [3,3,2',1'], and [4,3',2'] (the i's are marked).
Triangle starts:
  1;
  2;
  2,1;
  4,1;
  4,3;
  8,2,1;
  8,6,1;
From _Gus Wiseman_, Jul 11 2025: (Start)
Row n = 8 counts the following partitions by number of singleton parts other than the largest part:
  (8)                (5,3)        (4,3,1)
  (4,4)              (6,2)        (5,2,1)
  (4,2,2)            (7,1)
  (6,1,1)            (3,3,2)
  (2,2,2,2)          (3,2,2,1)
  (3,3,1,1)          (4,2,1,1)
  (5,1,1,1)          (3,2,1,1,1)
  (2,2,2,1,1)
  (4,1,1,1,1)
  (2,2,1,1,1,1)
  (3,1,1,1,1,1)
  (2,1,1,1,1,1,1)
  (1,1,1,1,1,1,1,1)
(End)

Links

Crossrefs

Row sums are A000041.
Row lengths are A003056.
For distinct parts instead of anti-runs we have A116608.
Column k = 1 is A116931.
For runs instead of anti-runs we have A384881.
The strict case is A384905.
The corresponding rank statistic is A356228, non-strict version A384906.
The proper case is A385814, runs A385815.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
Cf. A000009, A008284, A024786, A047993, A098859, A116674, A287170, A325325, A384882, A385213.

Programs

  • Maple
    g := add(x^j*mul(1+t*x^i+x^(2*i)/(1-x^i), i = 1 .. j-1)/(1-x^j), j = 1 .. 80): gser := simplify(series(g, x = 0, 27)): for n from 0 to 25 do P[n] := sort(coeff(gser, x, n)) end do: for n to 25 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i, t) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, i-1, t or j>0)*
      `if`(t and j=1, x, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, false)):
seq(T(n), n=1..20);  # Alois P. Heinz, Feb 13 2016
  • Mathematica
    b[n_, i_, t_] := b[n, i, t] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, t || j > 0]*If[t && j == 1, x, 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, False]]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Length[Split[#,#1!=#2+1&]]==k&]],{n,0,10},{k,0,n}] (* Delete zeros for A268193. Gus Wiseman, Jul 10 2025 *)

Formula

T(n,0) = A116931(n).
Sum_{k>=1} T(n, k) = A000041(n) (the partition numbers).
Sum_{k>=1} k*T(n,k) = A024786(n-1).
G.f.: G(t,x) = Sum_{j>=1} ((x^j/(1-x^j))*Product_{i=1..j-1} (1 + tx^i + x^{2i}/(1-x^i))).

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

Views

Author

Emeric Deutsch, Feb 22 2006

Keywords

Comments

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

Examples

			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)

Links

Crossrefs

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.

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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

A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 1, 0, 0, 1, 10, 5, 0, 0, 0, 1, 15, 15, 1, 0, 0, 0, 1, 21, 35, 7, 0, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 1, 36, 126, 84, 9, 0, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Philippe Deléham, Dec 10 2011

Keywords

Comments

Riordan array (1/(1-x), x^2/(1-x)^2).
A skewed version of triangular array A085478.
Mirror image of triangle in A098158.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n),A087455(n), A146559(n), A000012(n), A011782(n), A001333(n),A026150(n), A046717(n), A084057(n), A002533(n), A083098(n),A084058(n), A003665(n), A002535(n), A133294(n), A090042(n),A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
From Gus Wiseman, Jul 08 2025: (Start)
After the first row this is also the number of subsets of {1..n-1} with k maximal runs (sequences of consecutive elements increasing by 1) for k = 0..n. For example, row n = 5 counts the following subsets:
{} {1} {1,3} . . .
{2} {1,4}
{3} {2,4}
{4} {1,2,4}
{1,2} {1,3,4}
{2,3}
{3,4}
{1,2,3}
{2,3,4}
{1,2,3,4}
Requiring n-1 gives A202064.
For anti-runs instead of runs we have A384893.
(End)

Examples

			Triangle begins :
1
1, 0
1, 1, 0
1, 3, 0, 0
1, 6, 1, 0, 0
1, 10, 5, 0, 0, 0
1, 15, 15, 1, 0, 0, 0
1, 21, 35, 7, 0, 0, 0, 0
1, 28, 70, 28, 1, 0, 0, 0, 0

Crossrefs

Column k = 1 is A000217.
Column k = 2 is A000332.
Row sums are A011782 (or A000079 shifted right).
Removing all zeros gives A034839 (requiring n-1 A034867).
Last nonzero term in each row appears to be A093178, requiring n-1 A124625.
Reversing rows gives A098158, without zeros A109446.
Without the k = 0 column we get A210039.
Row maxima appear to be A214282.
A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
A268193 counts integer partitions by number of maximal runs, for anti-runs A384881.
Cf. A000045, A000071, A007318, A010027, A053538, A084938, A202064, A208342, A210034, A384175, A384893.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n-1]],Length[Split[#,#2==#1+1&]]==k&]],{n,0,10},{k,0,n}] (* Gus Wiseman, Jul 08 2025 *)

Formula

T(n,k) = binomial(n,2k).
G.f.: (1-x)/((1-x)^2-y*x^2).
T(n,k)= Sum_{j, j>=0} T(n-1-j,k-1)*j with T(n,0)=1 and T(n,k)= 0 if k<0 or if n
T(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k) for n>1, T(0,0) = T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Nov 10 2013

A385572 Number of subsets of {1..n} with the same number of maximal runs (increasing by 1) as maximal anti-runs (increasing by more than 1).

Original entry on oeis.org

1, 2, 3, 4, 7, 12, 19, 34, 63, 112, 207, 394, 739, 1398, 2687, 5152, 9891, 19128, 37039, 71754, 139459, 271522, 528999, 1032308, 2017291, 3945186, 7723203, 15134440, 29679407, 58245068, 114389683, 224796210, 442021743, 869658304, 1711914351, 3371515306
Offset: 0

Views

Author

Gus Wiseman, Jul 04 2025

Keywords

Comments

Also the number of subsets of {1..n} with the same number of adjacent elements increasing by 1 as adjacent elements increasing by more than 1.

Examples

			The set {2,3,5,6,8} has maximal runs ((2,3),(5,6),(8)) and maximal anti-runs ((2),(3,5),(6,8)) so is counted under a(8).
The a(0) = 1 through a(6) = 19 subsets:
  {}  {}   {}   {}   {}       {}       {}
      {1}  {1}  {1}  {1}      {1}      {1}
           {2}  {2}  {2}      {2}      {2}
                {3}  {3}      {3}      {3}
                     {4}      {4}      {4}
                     {1,2,4}  {5}      {5}
                     {1,3,4}  {1,2,4}  {6}
                              {1,2,5}  {1,2,4}
                              {1,3,4}  {1,2,5}
                              {1,4,5}  {1,2,6}
                              {2,3,5}  {1,3,4}
                              {2,4,5}  {1,4,5}
                                       {1,5,6}
                                       {2,3,5}
                                       {2,3,6}
                                       {2,4,5}
                                       {2,5,6}
                                       {3,4,6}
                                       {3,5,6}

Links

Crossrefs

The LHS is counted by A034839 (for partitions A384881, strict A116674), rank statistic A069010.
The case containing n + 1 is A217615.
The RHS is counted by A384893 or A210034 (for partitions A268193, strict A384905), rank statistic A384890.
Subsets of this type are ranked by A385575.
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. A000071, A000079, A001629, A010027, A053538, A384177, A384879, A384889.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<5, [1, 2, 3, 4, 7][n+1], ((3*n-4)*a(n-1)-
      (3*n-5)*a(n-2)+(5*n-12)*a(n-3)-2*(4*n-11)*a(n-4)+4*(n-3)*a(n-5))/(n-1))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jul 06 2025
  • Mathematica
    Table[Length[Select[Subsets[Range[n]],Length[Split[#,#2==#1+1&]]==Length[Split[#,#2!=#1+1&]]&]],{n,0,10}]
  • PARI
    a(n)=polcoef([1,1,1]*[x,0,0;x,x^2,1;0,x,x]^n*[1,0,0]~,n) \\ Christian Sievers, Jul 06 2025

Formula

Let M be the matrix [1,0,0; 1,x,1/x; 0,1,1]. Then a(n) is the sum of the constant terms of the entries in the left column of M^n. - Christian Sievers, Jul 06 2025

Extensions

a(21) and beyond from Christian Sievers, Jul 06 2025

A385575 Numbers whose binary indices have the same number of adjacent parts differing by 1 as adjacent parts differing by more than 1.

Original entry on oeis.org

1, 2, 4, 8, 11, 13, 16, 19, 22, 25, 26, 32, 35, 38, 44, 49, 50, 52, 64, 67, 70, 76, 87, 88, 91, 93, 97, 98, 100, 104, 107, 109, 117, 128, 131, 134, 140, 151, 152, 155, 157, 167, 174, 176, 179, 182, 185, 186, 193, 194, 196, 200, 203, 205, 208, 211, 214, 217
Offset: 1

Views

Author

Gus Wiseman, Jul 04 2025

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Examples

			The terms together with their binary expansions and binary indices begin:
    1:       1 ~ {1}
    2:      10 ~ {2}
    4:     100 ~ {3}
    8:    1000 ~ {4}
   11:    1011 ~ {1,2,4}
   13:    1101 ~ {1,3,4}
   16:   10000 ~ {5}
   19:   10011 ~ {1,2,5}
   22:   10110 ~ {2,3,5}
   25:   11001 ~ {1,4,5}
   26:   11010 ~ {2,4,5}
   32:  100000 ~ {6}
   35:  100011 ~ {1,2,6}
   38:  100110 ~ {2,3,6}
   44:  101100 ~ {3,4,6}
   49:  110001 ~ {1,5,6}
   50:  110010 ~ {2,5,6}
   52:  110100 ~ {3,5,6}
   64: 1000000 ~ {7}
   67: 1000011 ~ {1,2,7}
   70: 1000110 ~ {2,3,7}
   76: 1001100 ~ {3,4,7}
   87: 1010111 ~ {1,2,3,5,7}
   88: 1011000 ~ {4,5,7}
   91: 1011011 ~ {1,2,4,5,7}
   93: 1011101 ~ {1,3,4,5,7}
   97: 1100001 ~ {1,6,7}
   98: 1100010 ~ {2,6,7}
  100: 1100100 ~ {3,6,7}

Crossrefs

The LHS rank statistic is A069010, counted by A034839 (for partitions A384881, A116674).
The RHS rank statistic is A384890, counted by A384893 (for partitions A268193, A384905).
Subsets of this type are counted by A385572, with n A217615.
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. A000079, A010027, A053538, A210034, A384177, A384879, A384889.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],Length[Split[bpe[#],#2==#1+1&]]==Length[Split[bpe[#],#2!=#1+1&]]&]
  • PARI
    is_ok(n)=hammingweight(n)==2*hammingweight(bitand(n,n>>1))+1 \\ Christian Sievers, Jul 18 2025

A385574 Number of integer partitions of n with the same number of adjacent equal parts as adjacent unequal parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 5, 6, 10, 11, 13, 17, 20, 30, 36, 44, 55, 70, 86, 98, 128, 156, 190, 235, 288, 351, 409, 499, 603, 722, 863, 1025, 1227, 1461, 1757, 2061, 2444, 2892, 3406, 3996, 4708, 5497, 6430, 7595, 8835, 10294, 12027, 13971, 16252, 18887, 21878
Offset: 0

Views

Author

Gus Wiseman, Jul 04 2025

Keywords

Comments

These are also integer partitions of n with the same number of distinct parts as maximal anti-runs of parts.

Examples

			The partition (5,3,2,1,1,1,1) has 4 runs ((5),(3),(2),(1,1,1,1)) and 4 anti-runs ((5,3,2,1),(1),(1),(1)) so is counted under a(14).
The a(1) = 1 through a(10) = 10 reversed partitions (A = 10):
  (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)      (9)      (A)
                 (112)  (113)  (114)  (115)  (116)    (117)    (118)
                        (122)         (133)  (224)    (144)    (226)
                                      (223)  (233)    (225)    (244)
                                             (11123)  (11124)  (334)
                                                      (11223)  (11125)
                                                               (11134)
                                                               (11224)
                                                               (11233)
                                                               (12223)

Links

Crossrefs

The RHS is counted by A116608, rank statistic A297155.
The LHS is counted by A133121, rank statistic A046660.
For related inequalities see A212165, A212168, A361204.
For subsets instead of partitions see A217615, A385572, A385575.
These partitions are ranked by A385576.
A000041 counts integer partitions, strict A000009.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A034839 counts subsets by number maximal runs, for partitions A384881, strict A116674.
A047993 counts partitions with max part = length (A106529).
A098859 counts Wilf partitions (complement A336866), compositions A242882.
A268193 counts partitions by maximal anti-runs, strict A384905, subsets A384893.
A355394 counts partitions with neighbors, complement A356236.
Cf. A000071, A003114, A008284, A010027, A047966, A210034, A325324, A325325, A356606, A384882, A384885.

Programs

  • Mathematica
    Table[Length[Select[Reverse/@IntegerPartitions[n],Length[Union[#]]==Length[Split[#,#2!=#1&]]&]],{n,0,30}]
  • PARI
    lista(n)=Vec(polcoef((prod(i=1,n,1+x^i/(t*(1-t*x^i))+O(x*x^n))-1)*t+1,0,t)) \\ Christian Sievers, Jul 18 2025

Formula

For a partition p, let s(p) be its sum, e(p) the number of equal adjacent pairs, and d(p) the number of distinct adjacent pairs. Then Sum_{p partition} x^s(p) * t^(e(p)-d(p)) = (Product_{i>=1} (1 + x^i/(t*(1-t*x^i))) - 1) * t + 1, so a(n) is the coefficient of x^n*t^0 of this expression. - Christian Sievers, Jul 18 2025

A385814 Triangle read by rows where T(n,k) is the number of integer partitions of n with k maximal proper anti-runs (sequences decreasing by more than 1).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 0, 2, 2, 1, 1, 1, 0, 3, 2, 3, 1, 1, 1, 0, 3, 4, 2, 3, 1, 1, 1, 0, 4, 5, 4, 3, 3, 1, 1, 1, 0, 5, 5, 6, 5, 3, 3, 1, 1, 1, 0, 6, 8, 7, 6, 6, 3, 3, 1, 1, 1, 0, 7, 9, 10, 8, 7, 6, 3, 3, 1, 1, 1
Offset: 0

Views

Author

Gus Wiseman, Jul 09 2025

Keywords

Examples

			The partition (8,5,4,2,1) has maximal proper anti-runs ((8,5),(4,2),(1)) so is counted under T(20,3).
The partition (8,5,3,2,2) has maximal proper anti-runs ((8,5,3),(2),(2)) so is also counted under T(20,3).
Row n = 8 counts the following partitions:
  .  8   611  5111  41111  32111   221111  2111111  11111111
     71  521  4211  3221   311111
     62  44   332   2222   22211
     53  431  3311
         422
Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  2  1  1  1
  0  2  2  1  1  1
  0  3  2  3  1  1  1
  0  3  4  2  3  1  1  1
  0  4  5  4  3  3  1  1  1
  0  5  5  6  5  3  3  1  1  1
  0  6  8  7  6  6  3  3  1  1  1
  0  7  9 10  8  7  6  3  3  1  1  1
  0  9 11 13 12  9  8  6  3  3  1  1  1
  0 10 14 16 15 13 10  8  6  3  3  1  1  1
  0 12 19 18 21 17 14 11  8  6  3  3  1  1  1
  0 14 21 26 23 24 19 15 11  8  6  3  3  1  1  1
  0 17 26 31 33 28 26 20 16 11  8  6  3  3  1  1  1
  0 19 32 37 40 39 31 28 21 16 11  8  6  3  3  1  1  1
  0 23 38 47 50 47 45 34 29 22 16 11  8  6  3  3  1  1  1
  0 26 45 57 61 61 54 48 36 30 22 16 11  8  6  3  3  1  1  1
  0 31 53 71 75 76 70 60 51 37 31 22 16 11  8  6  3  3  1  1  1

Crossrefs

Row sums are A000041, strict A000009.
Column k = 1 is A003114.
For anti-runs instead of proper anti-runs we have A268193.
The corresponding rank statistic is A356228.
For proper runs instead of proper anti-runs we have A384881.
For subsets instead of partitions we have A384893, runs A034839.
The strict case is A384905.
For runs instead of proper anti-runs we have A385815.
A007690 counts partitions with no singletons (ranks A001694), complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A047993 counts partitions with max part = length, ranks A106529.
A098859 counts Wilf partitions, complement A336866 (ranks A325992).
A116608 counts partitions by distinct parts.
A116931 counts sparse partitions, ranks A319630.
Cf. A001227, A008284, A089259, A116674, A239455, A325325, A356226, A384880, A384885, A384887, A384906.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[Split[#,#1>#2+1&]]==k&]],{n,0,10},{k,0,n}]

A385815 Triangle read by rows where T(n,k) is the number of integer partitions of n with k maximal runs of consecutive elements decreasing by 0 or 1.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 3, 0, 0, 0, 4, 1, 0, 0, 0, 5, 2, 0, 0, 0, 0, 7, 4, 0, 0, 0, 0, 0, 8, 7, 0, 0, 0, 0, 0, 0, 10, 12, 0, 0, 0, 0, 0, 0, 0, 13, 16, 1, 0, 0, 0, 0, 0, 0, 0, 15, 25, 2, 0, 0, 0, 0, 0, 0, 0, 0, 18, 34, 4, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Gus Wiseman, Jul 09 2025

Keywords

Examples

			The partition (8,5,4,2,1) has maximal runs ((8),(5,4),(2,1)) so is counted under T(20,3).
The partition (8,5,3,2,2) has maximal runs ((8),(5),(3,2,2)) so is also counted under T(20,3).
Row n = 9 counts the following partitions:
  (9)                  (6,3)            (5,3,1)
  (5,4)                (7,2)
  (3,3,3)              (8,1)
  (4,3,2)              (4,4,1)
  (3,2,2,2)            (5,2,2)
  (3,3,2,1)            (6,2,1)
  (2,2,2,2,1)          (7,1,1)
  (3,2,2,1,1)          (4,2,2,1)
  (2,2,2,1,1,1)        (4,3,1,1)
  (3,2,1,1,1,1)        (5,2,1,1)
  (2,2,1,1,1,1,1)      (6,1,1,1)
  (2,1,1,1,1,1,1,1)    (3,3,1,1,1)
  (1,1,1,1,1,1,1,1,1)  (4,2,1,1,1)
                       (5,1,1,1,1)
                       (4,1,1,1,1,1)
                       (3,1,1,1,1,1,1)
Triangle begins:
   1
   0   1
   0   2   0
   0   3   0   0
   0   4   1   0   0
   0   5   2   0   0   0
   0   7   4   0   0   0   0
   0   8   7   0   0   0   0   0
   0  10  12   0   0   0   0   0   0
   0  13  16   1   0   0   0   0   0   0
   0  15  25   2   0   0   0   0   0   0   0
   0  18  34   4   0   0   0   0   0   0   0   0
   0  23  46   8   0   0   0   0   0   0   0   0   0
   0  26  62  13   0   0   0   0   0   0   0   0   0   0
   0  31  82  22   0   0   0   0   0   0   0   0   0   0   0

Crossrefs

Row sums are A000041, strict A000009.
Column k = 1 is A034296 (flat or gapless partitions, ranks A066311 or A073491).
For subsets instead of partitions we have A034839, anti-runs A384893.
The strict case appears to be A116674.
For anti-runs instead of runs we have A268193.
The corresponding rank statistic is A287170.
For proper runs instead of runs we have A384881.
For proper anti-runs instead of runs we have A385814.
A007690 counts partitions with no singletons (ranks A001694), complement A183558.
A047993 counts partitions with max part = length, rank A106529.
A098859 counts Wilf partitions, complement A336866 (ranks A325992).
A116608 counts partitions by distinct parts.
A116931 counts sparse partitions, ranks A319630.
Cf. A001227, A008284, A325325, A356226, A384884, A384885, A384887, A384905.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[Split[#,#1<=#2+1&]]==k&]],{n,0,20},{k,0,n}]
Showing 1-8 of 8 results.

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