A002846 -id:A002846 - cp's OEIS frontend

A327699 Number of refinement sequences n -> ... -> {1}^n, where in each step every single part of a nonempty selection of parts is replaced by a partition of itself into two smaller parts (in weakly decreasing order).

1, 1, 1, 4, 9, 48, 211, 1736, 9777, 91169, 739174, 8613817, 83763730, 1105436491, 13222076337, 207852246589, 2789691577561, 47759515531854, 755158220565169, 14595210284816038, 255814560447492788, 5373613110108953192, 105867623217924984398, 2460702471446564481641

Offset: 1

Alois P. Heinz, Sep 22 2019

nonn

			a(4) = 4:
  4 -> 31   -> 211  -> 1111
  4 -> 22   -> 1111
  4 -> 22   -> 112  -> 1111
  4 -> 22   -> 211  -> 1111

Wikipedia, Partition (number theory)

Cf. A002846, A327643, A327697, A327698, A327702.

A276027 Number of ways to transform a sequence of n ones to a single number by continually removing two numbers and replacing them with their sum modulo 3.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 18, 43, 93, 266, 702, 1687, 5136, 14405, 36898, 117016, 341842, 914064, 2983027, 8972121, 24743851, 82478973, 253555061, 715745648, 2424954125, 7582390623, 21796481477, 74805170349, 237095926682, 691568408221, 2398418942361, 7686495623620

Offset: 1

Caleb Ji, Aug 16 2016

nonn

Can be considered as the number of maximal chains in a poset whose nodes are the possible states of the sequence. In this sense it counts the same things as A002846 when the elements of that poset are taken modulo 3.

Originally this entry had a reference to a paper on the arXiv by Caleb Ji, Enumerative Properties of Posets Corresponding to a Certain Class of No Strategy Games, arXiv:1608.06025 [math.CO], 2016. However, this article has since been removed from the arXiv. - N. J. A. Sloane, Sep 07 2018

			For n = 4, the two ways are 1111 -> 211 -> 10 -> 1 and 1111 -> 211 -> 22 -> 1.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
C. Ji, Example for a(7)

Cf. A002846, A117143.
Similar to A002846 with nodes taken modulo 3.
A117143 is the total number of nodes in this poset.

b:= proc(x, y, z) option remember;
      `if`(x+y+z=1, 1, `if`(y>0 and z>0, b(x+1, y-1, z-1), 0)+
      `if`(x>1 or x>0 and y>0 or x>0 and z>0, b(x-1, y, z), 0)+
      `if`(y>1, b(x, y-2, z+1), 0)+`if`(z>1, b(x, y+1, z-2), 0))
    end:
a:= n-> b(0, n, 0):
seq(a(n), n=1..35);  # Alois P. Heinz, Aug 18 2016

b[x_,y_,z_] := b[x, y, z] = If[x+y+z==1, 1,  If[y>0 && z>0, b[x+1, y-1, z-1], 0] + If[x>1 || x>0 && y>0 || x>0 && z>0, b[x-1, y, z], 0] + If[y>1, b[x, y-2, z+1], 0] + If[z>1, b[x, y+1, z-2], 0]]; a[n_]:= b[0, n, 0]; Array[a,35] (* Jean-François Alcover, Aug 07 2017, after Alois P. Heinz *)

from sympy.core.cache import cacheit
@cacheit
def b(x, y, z): return 1 if x + y + z==1 else (b(x + 1, y - 1, z - 1) if y>0 and z>0 else 0) + (b(x - 1, y, z) if x>1 or x>0 and y>0 or x>0 and z>0 else 0) + (b(x, y - 2, z + 1) if y>1 else 0) + (b(x, y + 1, z - 2) if z>1 else 0)
def a(n): return b(0, n, 0)
print([a(n) for n in range(1, 36)]) # Indranil Ghosh, Aug 09 2017, after Maple code

a(n) = f(0, n, 0) where f(a, b, c) is the number of ways to reach one number beginning with a zeros, b ones, and c twos.

Then f(a, b, c) = f_1 + f_2 + f_3 + f_4 where f_1 = f(a-1, b, c) if a>=2 or a, b >=1 or a,c >=1, f_2 = f(a, b-2, c+1) if b >= 2, f_3 = f(a, b+1, c-2) if c >= 2, and f_4 = f(a+1, b-1, c-1) if b, c >= 1, and each are 0 otherwise. Initial terms: f(a, b, c) = 1 for all 1 <= a+b+c <= 2, where a, b, c >= 0.

a(19)-a(32) from Alois P. Heinz, Aug 18 2016

A330784 Triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k with n equal atoms.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 9, 5, 1, 9, 28, 36, 16, 1, 13, 69, 160, 164, 61, 1, 20, 160, 580, 1022, 855, 272, 1, 28, 337, 1837, 4996, 7072, 4988, 1385

Offset: 2

Gus Wiseman, Jan 03 2020

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

			Triangle begins:
    1
    1    1
    1    3    2
    1    5    9    5
    1    9   28   36   16
    1   13   69  160  164   61
    1   20  160  580 1022  855  272
    1   28  337 1837 4996 7072 4988 1385
Row n = 5 counts the following multisystems (strings of 1's are replaced by their lengths):
  5  {1,4}      {{1},{1,3}}      {{{1}},{{1},{1,2}}}
     {2,3}      {{1},{2,2}}      {{{1,1}},{{1},{2}}}
     {1,1,3}    {{2},{1,2}}      {{{1}},{{2},{1,1}}}
     {1,2,2}    {{3},{1,1}}      {{{1,2}},{{1},{1}}}
     {1,1,1,2}  {{1},{1,1,2}}    {{{2}},{{1},{1,1}}}
                {{1,1},{1,2}}
                {{2},{1,1,1}}
                {{1},{1},{1,2}}
                {{1},{2},{1,1}}

Row sums are A318813.
Column k = 3 is A007042.
Column k = 4 is A001970(n) - 3*A000041(n) + 3.
Column k = n is A000111.
Row n is row prime(n) of A330727.
Cf. A000669, A001055, A002846, A005121, A196545, A213427, A318812, A320160, A330474, A330475, A330655, A330667, A330679.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

T(n,3) = A000041(n) - 2.

T(n,4) = A001970(n) - 3 * A000041(n) + 3.

A213597 Triangle T(n,k), n>=1, 0<=k<=A000041(n), read by rows: row n gives the coefficients of the chromatic polynomial of the ranked poset L(n) of partitions of n, highest powers first.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -5, 10, -9, 3, 0, 1, -9, 36, -79, 98, -64, 17, 0, 1, -17, 136, -666, 2192, -5032, 8111, -9013, 6569, -2818, 537, 0, 1, -28, 378, -3242, 19648, -88676, 306308, -819933, 1703404, -2723374, 3285552, -2887734, 1739326, -639065, 107435, 0

Offset: 1

Alois P. Heinz, Jun 15 2012

The ranked poset L(n) of partitions is defined in A002846. A partition of n into k parts is connected to another partition of n into k+1 parts that results from splitting one part of the first partition into two parts.

			L(5):     (32)---(221)
         /    \ /     \
        /      X       \
       /      / \       \
    (5)---(41)---(311)---(2111)---(11111)
Chromatic polynomial: q^7-9*q^6+36*q^5-79*q^4+98*q^3-64*q^2+17*q.
Triangle T(n,k) begins:
  1,   0;
  1,  -1,   0;
  1,  -2,   1,    0;
  1,  -5,  10,   -9,    3,     0;
  1,  -9,  36,  -79,   98,   -64,   17,     0;
  1, -17, 136, -666, 2192, -5032, 8111, -9013, 6569, -2818, 537, 0;

Alois P. Heinz, Rows n = 1..9, flattened
Olivier Gérard, The ranked posets L(2),...,L(8)
Eric Weisstein's World of Mathematics, Chromatic Polynomial
Wikipedia, Chromatic Polynomial

Row lengths give: 1+A000041(n) = A052810(n).
Row sums (for n>1) and last elements of rows give: A000004.
Columns k=1-2 give: A000012, (-1)*A000097(n-2).
Cf. A002846, A213242, A213385, A213427.

Edited by Alois P. Heinz at the suggestion of Gus Wiseman, May 02 2016

A330726 Number of balanced reduced multisystems of maximum depth whose atoms are positive integers summing to n.

Original entry on oeis.org

1, 1, 2, 3, 7, 17, 54, 199, 869, 4341, 24514, 154187

Offset: 0

Gus Wiseman, Jan 03 2020

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

			The a(1) = 1 through a(5) = 17 multisystems (commas elided):
  {1}  {2}   {3}        {4}               {5}
       {11}  {12}       {22}              {23}
             {{1}{11}}  {13}              {14}
                        {{1}{12}}         {{1}{13}}
                        {{2}{11}}         {{1}{22}}
                        {{{1}}{{1}{11}}}  {{2}{12}}
                        {{{11}}{{1}{1}}}  {{3}{11}}
                                          {{{1}}{{1}{12}}}
                                          {{{11}}{{1}{2}}}
                                          {{{1}}{{2}{11}}}
                                          {{{12}}{{1}{1}}}
                                          {{{2}}{{1}{11}}}
                                          {{{{1}}}{{{1}}{{1}{11}}}}
                                          {{{{1}}}{{{11}}{{1}{1}}}}
                                          {{{{1}{1}}}{{{1}}{{11}}}}
                                          {{{{1}{11}}}{{{1}}{{1}}}}
                                          {{{{11}}}{{{1}}{{1}{1}}}}

The case with all atoms equal to 1 is A000111.
The non-maximal version is A330679.
A tree version is A320160.
Cf. A000669, A002846, A005121, A141268, A196545, A213427, A317145, A318813, A330663, A330665, A330675, A330676, A330728.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

A276032 Number of refinements of the partition n^1 with all numbers taken modulo 2.

Original entry on oeis.org

1, 2, 3, 7, 8, 21, 22, 63, 64, 195, 196, 624, 625, 2054, 2055, 6916, 6917, 23712, 23713, 82498, 82499, 290510, 290511, 1033410, 1033411, 3707850, 3707851, 13402695, 13402696, 48760365, 48760366, 178405155, 178405156, 656043855, 656043856, 2423307045

Offset: 1

Caleb Ji, Aug 17 2016

nonn

Consider the ranked poset L(n) of partitions defined in A002846, and take the elements of each node modulo 2, collapsing two equivalent nodes into 1. Then a(n) is the total number of paths of all lengths 0,1,...,n-1 that start at (n mod 2)^1 and end at any node in the poset.
Odd-indexed terms are the partial sums of Catalan numbers: A014138.
Even-indexed terms are one less than the following odd-indexed term.
Originally this entry had a reference to a paper on the arXiv by Caleb Ji, Enumerative Properties of Posets Corresponding to a Certain Class of No Strategy Games, arXiv:1608.06025 [math.CO], 2016. However, this article has since been removed from the arXiv. - N. J. A. Sloane, Sep 07 2018

Cf. A276033, A014138.

A276033 Number of generalizations of the partition 1^n with all elements taken modulo 2.

Original entry on oeis.org

1, 2, 3, 6, 8, 17, 22, 50, 64, 154, 196, 493, 625, 1626, 2055, 5487, 6917, 18851, 23713, 65703, 82499, 231725, 290511, 825399, 1033411, 2964951, 3707851, 10728256, 13402696, 39065521, 48760366, 143047486, 178405156, 526399066, 656043856, 1945668346, 2423307046

Offset: 1

Caleb Ji, Aug 17 2016

nonn

Consider the ranked poset L(n) of partitions defined in A002846, and take the elements of each node modulo 2, collapsing two equivalent nodes into 1. Then a(n) is the total number of paths of all lengths 0,1,...,n-1 that start at any node in the poset and end at 1^n.
Odd-indexed terms are the partial sums of Catalan numbers: A014138.  Even-indexed terms a_{2n} are C_n (the n-th Catalan number) less than the following odd-indexed term.
Originally this entry had a reference to a paper on the arXiv by Caleb Ji, Enumerative Properties of Posets Corresponding to a Certain Class of No Strategy Games, arXiv:1608.06025 [math.CO], 2016. However, this article has since been removed from the arXiv. - N. J. A. Sloane, Sep 07 2018

Cf. A276032, A014138.

A300384 In the ranked poset of integer partitions ordered by refinement, number of maximal chains from the local minimum to the partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 11, 2, 2, 1, 33, 1, 116, 1, 5, 4, 435, 1, 2, 11, 1, 2, 1832, 2, 8167, 1, 12, 33, 10, 1, 39700, 116, 37, 1, 201785, 5, 1099449, 4, 3, 435, 6237505, 1, 19, 2, 123, 11, 37406458, 1, 27, 2, 474, 1832, 232176847, 2, 1513796040

Offset: 1

Gus Wiseman, Mar 04 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(21) = 5 maximal chains are the rows:
(111111)<(21111)<(2211)<(222)<(42)
(111111)<(21111)<(2211)<(411)<(42)
(111111)<(21111)<(2211)<(321)<(42)
(111111)<(21111)<(3111)<(411)<(42)
(111111)<(21111)<(3111)<(321)<(42)

Cf. A000041, A001055, A001222, A002846, A056239, A112798, A213427, A215366, A265947, A296150, A299200, A299202, A299925, A300273, A300383, A300385.

pcovs[ptn_]:=Select[Union[Reverse/@Sort/@Join@@@Tuples[IntegerPartitions/@ptn]],Length[#]===Length[ptn]+1&];
coc[ptn_]:=coc[ptn]=If[Max[ptn]===1,1,Total[coc/@pcovs[ptn]]];
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[coc[Reverse[primeMS[n]]],{n,50}]

A330785 Triangle read by rows where T(n,k) is the number of chains of length k from minimum to maximum in the poset of integer partitions of n ordered by refinement.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 5, 8, 4, 0, 1, 9, 25, 28, 11, 0, 1, 13, 57, 111, 99, 33, 0, 1, 20, 129, 379, 561, 408, 116, 0, 1, 28, 253, 1057, 2332, 2805, 1739, 435, 0, 1, 40, 496, 2833, 8695, 15271, 15373, 8253, 1832, 0, 1, 54, 898, 6824, 28071, 67790, 98946, 85870, 40789, 8167

Offset: 1

Gus Wiseman, Jan 03 2020

			Triangle begins:
   1
   0   1
   0   1   1
   0   1   3   2
   0   1   5   8   4
   0   1   9  25  28  11
   0   1  13  57 111  99  33
   0   1  20 129 379 561 408 116
Row n = 5 counts the following chains (minimum and maximum not shown):
  ()  (14)    (113)->(14)    (1112)->(113)->(14)
      (23)    (113)->(23)    (1112)->(113)->(23)
      (113)   (122)->(14)    (1112)->(122)->(14)
      (122)   (122)->(23)    (1112)->(122)->(23)
      (1112)  (1112)->(14)
              (1112)->(23)
              (1112)->(113)
              (1112)->(122)

Row sums are A213427.
Main diagonal is A002846.
Column k=3 is A007042.
Dominated by A330784.
The version for set partitions is A008826.
The version for factorizations is A330935.
Cf. A000111, A000258, A000311, A005121, A141268, A196545, A265947, A300383, A306186, A317141, A317176, A318813, A320160, A330679.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
upr[q_]:=Union[Sort/@Apply[Plus,mps[q],{2}]];
paths[eds_,start_,end_]:=If[start==end,Prepend[#,{}],#]&[Join@@Table[Prepend[#,e]&/@paths[eds,Last[e],end],{e,Select[eds,First[#]==start&]}]];
Table[Length[Select[paths[Join@@Table[{y,#}&/@DeleteCases[upr[y],y],{y,Sort/@IntegerPartitions[n]}],ConstantArray[1,n],{n}],Length[#]==k-1&]],{n,8},{k,n}]

T(n,k) = A330935(2^n,k).

cp's OEIS Frontend

A327699 Number of refinement sequences n -> ... -> {1}^n, where in each step every single part of a nonempty selection of parts is replaced by a partition of itself into two smaller parts (in weakly decreasing order).

Views

Author

Keywords

Examples

Links

Crossrefs

A276027 Number of ways to transform a sequence of n ones to a single number by continually removing two numbers and replacing them with their sum modulo 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

Extensions

A330784 Triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k with n equal atoms.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A213597 Triangle T(n,k), n>=1, 0<=k<=A000041(n), read by rows: row n gives the coefficients of the chromatic polynomial of the ranked poset L(n) of partitions of n, highest powers first.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A330726 Number of balanced reduced multisystems of maximum depth whose atoms are positive integers summing to n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A276032 Number of refinements of the partition n^1 with all numbers taken modulo 2.

Views

Author

Keywords

Comments

Crossrefs

A276033 Number of generalizations of the partition 1^n with all elements taken modulo 2.

Views

Author

Keywords

Comments

Crossrefs

A300384 In the ranked poset of integer partitions ordered by refinement, number of maximal chains from the local minimum to the partition with Heinz number n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A330785 Triangle read by rows where T(n,k) is the number of chains of length k from minimum to maximum in the poset of integer partitions of n ordered by refinement.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula