A114901 -id:A114901 - cp's OEIS frontend

A353430 Number of integer compositions of n that are empty, a singleton, or whose own run-lengths are a consecutive subsequence that is already counted.

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 5, 7, 9, 11, 15, 16, 22, 25, 37, 37, 45

Offset: 0

Gus Wiseman, May 16 2022

			The a(n) compositions for selected n (A..E = 10..14):
  n=4:  n=6:    n=9:      n=10:     n=12:     n=14:
-----------------------------------------------------------
  (4)   (6)     (9)       (A)       (C)       (E)
  (22)  (1122)  (333)     (2233)    (2244)    (2255)
        (2211)  (121122)  (3322)    (4422)    (5522)
                (221121)  (131122)  (151122)  (171122)
                          (221131)  (221124)  (221126)
                                    (221142)  (221135)
                                    (221151)  (221153)
                                    (241122)  (221162)
                                    (421122)  (221171)
                                              (261122)
                                              (351122)
                                              (531122)
                                              (621122)
                                              (122121122)
                                              (221121221)

Non-recursive non-consecutive version: counted by A353390, ranked by A353402, reverse A353403, partitions A325702.
Non-consecutive version: A353391, ranked by A353431, partitions A353426.
Non-recursive version: A353392, ranked by A353432.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A114901 counts compositions with no runs of length 1.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223.
A329738 counts uniform compositions, partitions A047966.
A329739 counts compositions with all distinct run-lengths.
Cf. A005811, A032020, A103295, A114640, A165413, A242882, A325705, A333755, A351013, A353400, A353401.

yoyQ[y_]:=Length[y]<=1||MemberQ[Join@@Table[Take[y,{i,j}],{i,Length[y]},{j,i,Length[y]}],Length/@Split[y]]&&yoyQ[Length/@Split[y]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],yoyQ]],{n,0,15}]

A373952 Number of integer compositions of n whose run-compression sums to 3.

Original entry on oeis.org

0, 0, 0, 3, 2, 4, 5, 6, 6, 9, 8, 10, 11, 12, 12, 15, 14, 16, 17, 18, 18, 21, 20, 22, 23, 24, 24, 27, 26, 28, 29, 30, 30, 33, 32, 34, 35, 36, 36, 39, 38, 40, 41, 42, 42, 45, 44, 46, 47, 48, 48, 51, 50, 52, 53, 54, 54, 57, 56, 58, 59, 60, 60, 63, 62, 64, 65, 66

Offset: 0

Gus Wiseman, Jun 29 2024

nonn

We define the (run-) compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has compression (1,2,1).

			The a(3) = 3 through a(9) = 9 compositions:
  (3)   (112)  (122)   (33)     (1222)    (11222)    (333)
  (12)  (211)  (221)   (1122)   (2221)    (22211)    (12222)
  (21)         (1112)  (2211)   (11122)   (111122)   (22221)
               (2111)  (11112)  (22111)   (221111)   (111222)
                       (21111)  (111112)  (1111112)  (222111)
                                (211111)  (2111111)  (1111122)
                                                     (2211111)
                                                     (11111112)
                                                     (21111111)

John Tyler Rascoe, Table of n, a(n) for n = 0..10000

For partitions we appear to have A137719.
Column k = 3 of A373949, rows-reversed A373951.
The compression-sum statistic is represented by A373953, difference A373954.
A003242 counts compressed compositions (anti-runs).
A011782 counts compositions.
A114901 counts compositions with no isolated parts.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A240085 counts compositions with no unique parts.
A333755 counts compositions by compressed length.
A373948 represents the run-compression transformation.
Cf. A037201 (halved A373947), A106356, A124762, A238130, A238279, A238343, A285981, A333213, A333489, A373950.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#]]==3&]],{n,0,10}]

A_x(N)={my(x='x+O('x^N)); concat([0, 0, 0], Vec(x^3 *(3-x-x^2-x^3)/((1-x)*(1-x^2)*(1-x^3))))}
A_x(50) \\ John Tyler Rascoe, Jul 01 2024

G.f.: x^3 * (3-x-x^2-x^3)/((1-x)*(1-x^2)*(1-x^3)). - John Tyler Rascoe, Jul 01 2024

a(26) onwards from John Tyler Rascoe, Jul 01 2024

A329864 Number of compositions of n with the same runs-resistance as cuts-resistance.

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 5, 10, 17, 27, 68, 107, 217, 420, 884, 1761, 3679, 7469, 15437, 31396, 64369

Offset: 0

Gus Wiseman, Nov 23 2019

A composition of n is a finite sequence of positive integers summing to n.
For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined to be the number of applications required to reach a singleton.
For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

			The a(5) = 2 through a(8) = 17 compositions:
  (1112)  (1113)   (1114)    (1115)
  (2111)  (1122)   (1222)    (1133)
          (2211)   (2221)    (3311)
          (3111)   (4111)    (5111)
          (11211)  (11122)   (11222)
                   (11311)   (11411)
                   (21112)   (12221)
                   (22111)   (21113)
                   (111121)  (22211)
                   (121111)  (31112)
                             (111131)
                             (111221)
                             (112112)
                             (112211)
                             (122111)
                             (131111)
                             (211211)
For example, the runs-resistance of (111221) is 3 because we have: (111221) -> (321) -> (111) -> (3), while the cuts-resistance is also 3 because we have: (111221) -> (112) -> (1) -> (), so (111221) is counted under a(8).

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

The version for binary expansion is A329865.
Compositions counted by runs-resistance are A329744.
Compositions counted by cuts-resistance are A329861.
Compositions with runs-resistance = cuts-resistance minus 1 are A329869.
Cf. A003242, A098504, A114901, A242882, A318928, A319411, A319416, A319420, A319421, A329867, A329868.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==degdep[#]&]],{n,0,10}]

A353427 Numbers k such that the k-th composition in standard order has all run-lengths > 1.

Original entry on oeis.org

0, 3, 7, 10, 15, 31, 36, 42, 43, 58, 63, 87, 122, 127, 136, 147, 170, 171, 175, 228, 234, 235, 250, 255, 292, 295, 343, 351, 471, 484, 490, 491, 506, 511, 528, 547, 586, 591, 676, 682, 683, 687, 698, 703, 904, 915, 938, 939, 943, 983, 996, 1002, 1003, 1018

Offset: 1

Gus Wiseman, May 16 2022

nonn

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.

			The terms and corresponding compositions begin:
     0: ()
     3: (1,1)
     7: (1,1,1)
    10: (2,2)
    15: (1,1,1,1)
    31: (1,1,1,1,1)
    36: (3,3)
    42: (2,2,2)
    43: (2,2,1,1)
    58: (1,1,2,2)
    63: (1,1,1,1,1,1)
    87: (2,2,1,1,1)
   122: (1,1,1,2,2)
   127: (1,1,1,1,1,1,1)

The version for partitions is A001694, counted by A007690.
The version for parts instead of lengths is A022340, counted by A212804.
These compositions are counted by A114901.
A subset of A348612 (counted by A261983).
The case of all run-lengths = 2 is A351011.
The case of all run-lengths > 2 is counted by A353400.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order, reverse A228351.
Statistics of standard compositions:
- Length is A000120, sum A070939.
- Runs are counted by A124767.
- Runs-resistance is A333628.
- Run-lengths are A333769.
Cf. A044813, A128695, A165413, A240085, A244164, A274174, A318928, A333489, A333755, A353402, A353432.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!MemberQ[Length/@Split[stc[#]],1]&]

A353428 Number of integer compositions of n with all parts and all run-lengths > 2.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 2, 4, 0, 0, 8, 3, 0, 10, 4, 4, 15, 4, 8, 24, 7, 8, 42, 16, 10, 59, 31, 27, 87, 37, 52, 149, 62, 66, 233, 121, 111, 342, 207, 204, 531, 308, 351, 864, 487, 536, 1373, 864, 865, 2057, 1440, 1509, 3232

Offset: 0

Gus Wiseman, May 16 2022

nonn

			The a(n) compositions for selected n:
  n=16:   n=18:     n=20:    n=21:      n=24:
----------------------------------------------------
  (4444)  (666)     (5555)   (777)      (888)
          (333333)  (44444)  (333444)   (6666)
                             (444333)   (333555)
                             (3333333)  (444444)
                                        (555333)
                                        (3333444)
                                        (4443333)
                                        (33333333)

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Allowing any multiplicities gives A078012, partitions A008483.
The version for no (instead of all) parts or run-lengths > 2 is A137200.
Allowing any parts gives A353400, partitions A100405.
The version for partitions is A353501, ranked by A353502.
The version for > 1 instead of > 2 is A353508, partitions A339222.
A003242 counts anti-run compositions, ranked by A333489.
A008466 counts compositions with some part > 2.
A011782 counts compositions.
A114901 counts compositions with no runs of length 1, ranked by A353427.
A128695 counts compositions with no run-lengths > 2.
A261983 counts non-anti-run compositions.
A335464 counts compositions with a run-length > 2.
Cf. A032020, A044813, A098859, A165413, A318928, A329738, A329739, A333755, A353390, A353429.

b:= proc(n, h) option remember; `if`(n=0, 1, add(
     `if`(i=h, 0, add(b(n-i*j, i), j=3..n/i)), i=3..n/3))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, May 18 2022

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[#,1|2]&&!MemberQ[Length/@Split[#],1|2]&]],{n,0,15}]

a(26)-a(66) from Alois P. Heinz, May 17 2022

A353502 Numbers with all prime indices and exponents > 2.

Original entry on oeis.org

1, 125, 343, 625, 1331, 2197, 2401, 3125, 4913, 6859, 12167, 14641, 15625, 16807, 24389, 28561, 29791, 42875, 50653, 68921, 78125, 79507, 83521, 103823, 117649, 130321, 148877, 161051, 166375, 205379, 214375, 226981, 274625, 279841, 300125, 300763, 357911

Offset: 1

Gus Wiseman, May 16 2022

nonn

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.

			The initial terms together with their prime indices:
       1: {}
     125: {3,3,3}
     343: {4,4,4}
     625: {3,3,3,3}
    1331: {5,5,5}
    2197: {6,6,6}
    2401: {4,4,4,4}
    3125: {3,3,3,3,3}
    4913: {7,7,7}
    6859: {8,8,8}
   12167: {9,9,9}
   14641: {5,5,5,5}
   15625: {3,3,3,3,3,3}
   16807: {4,4,4,4,4}
   24389: {10,10,10}
   28561: {6,6,6,6}
   29791: {11,11,11}
   42875: {3,3,3,4,4,4}

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 3000 terms from Amiram Eldar)

The version for only parts is A007310, counted by A008483.
The version for <= 2 instead of > 2 is A018256, # of compositions A137200.
The version for only multiplicities is A036966, counted by A100405.
The version for indices and exponents prime (instead of > 2) is:
- listed by A346068
- counted by A351982
- only exponents: A056166, counted by A055923
- only parts: A076610, counted by A000607
The version for > 1 instead of > 2 is A062739, counted by A339222.
The version for compositions is counted by A353428, see A078012, A353400.
The partitions with these Heinz numbers are counted by A353501.
A000726 counts partitions with multiplicities <= 2, compositions A128695.
A001222 counts prime factors with multiplicity, distinct A001221.
A004250 counts partitions with some part > 2, compositions A008466.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A295341 counts partitions with some multiplicity > 2, compositions A335464.
Cf. A000720, A003963, A065483, A114901, A181819, A353503, A353508.

Select[Range[10000],#==1||!MemberQ[FactorInteger[#],{?(#<5&),}|{,?(#<3&)}]&]

Sum_{n>=1} 1/a(n) = Product_{p prime > 3} (1 + 1/(p^2*(p-1))) = (72/95)*A065483 = 1.0154153584... . - Amiram Eldar, May 28 2022

A373950 Number of integer compositions of n containing two adjacent ones and no other runs.

Original entry on oeis.org

0, 0, 1, 0, 2, 4, 5, 14, 26, 46, 92, 176, 323, 610, 1145, 2108, 3912, 7240, 13289, 24418, 44778, 81814, 149356, 272222, 495144, 899554, 1632176, 2957332, 5352495, 9677266, 17477761, 31536288, 56852495, 102403134, 184302331, 331452440, 595659234, 1069742760

Offset: 0

Gus Wiseman, Jun 28 2024

nonn

Also the number of integer compositions of n such that replacing each run of repeated parts with a single part (run-compression) results in a composition of n-1.

			The a(0) = 0 through a(7) = 14 compositions:
  .  .  (11)  .  (112)  (113)   (114)   (115)
                 (211)  (311)   (411)   (511)
                        (1121)  (1131)  (1123)
                        (1211)  (1311)  (1132)
                                (2112)  (1141)
                                        (1411)
                                        (2113)
                                        (2311)
                                        (3112)
                                        (3211)
                                        (11212)
                                        (12112)
                                        (21121)
                                        (21211)

John Tyler Rascoe, Table of n, a(n) for n = 0..1000

For any run (not just of ones) we have A003242.
Subdiagonal of A373949.
These compositions are ranked by A373956.
A003242 counts compressed compositions.
A114901 counts compositions with no isolated parts.
A116861 counts partitions by compressed sum, by compressed length A116608.
A333755 counts compositions by compressed length (number of runs).
A373948 represents the run-compression transformation.
Cf. A106356, A238130, A238279, A238343, A240085, A285981, A333213, A333489, A373951, A373952.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Total[First/@Split[#]]==n-1&]],{n,0,10}]

A_x(N)={my(x='x+O('x^N), h=x/((1+x)^2*(1-sum(i=1,N, (x^i /(1+x^i))))^2)); concat([0, 0], Vec(h))}
A_x(40) \\ John Tyler Rascoe, Jul 02 2024

a(n>0) = A373949(n,n-1).

G.f.: x/((1-x)^2 * (1 - Sum_{i>0} (x^i/(1+x^i)))^2). - John Tyler Rascoe, Jul 02 2024

a(26) onwards from John Tyler Rascoe, Jul 02 2024

A273461 Number of physically stable n X n placements of water source-blocks in Minecraft.

Original entry on oeis.org

1, 2, 9, 40, 484, 9717, 338724, 21624680, 2504301849, 520443847520, 195145309791364, 131850659243316222, 160668896658179472676, 352891729183598844656996, 1397187513066371784602204416, 9972288382286063615850619475640

Offset: 0

Gus Wiseman, May 23 2016

nonn

In Minecraft worlds, a source block of water can be reacted with another source block, two blocks away. This reaction creates a third "infinite" source block in the unoccupied intermediate block, so called because if the intermediate water source is destroyed or picked up by a player using a bucket, it will immediately regenerate itself.
A placement of water at several positions in an n X n board is said to be *stable* if no infinite water physics can in fact occur (under otherwise optimal conditions). This means that the total quantity of water in the system is held constant.
In short, no two source blocks can be graph-distance 2 from each other. - Gus Wiseman, Nov 27 2019
Often incorrectly described as cellular automata, the observed behaviors of liquids within a board are inseparable in certain ways from states of affair outside of the board and events outside of the system. This aspect of Minecraft is poorly understood.

			a(2) = 9: {{}, {(2,2)}, {(2,1)}, {(2,1),(2,2)}, {(1,2)}, {(1,2),(2,2)}, {(1,1)}, {(1,1),(2,1)}, {(1,1),(1,2)}}.

EthosLab, Minecraft - Tutorial: Water
Wikipedia, Distance (graph theory)
Gus Wiseman, Example of a physically stable arrangement of water source-blocks (n=11)
Christopher Cormier, C# Program

The one-dimensional version is A006498.
Dominated by A329871.
Cf. A002416, A005251, A027624, A114901.

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
allflows[n_]:=stableSets[Join@@Array[List,{n,n}],Function[{v,w},Plus@@Abs/@(w-v)===2]];
Table[Length[allflows[i]],{i,6}] (* Gus Wiseman, May 23 2016 *)

a(7) from Tae Lim Kook, May 25 2016
a(8) from Tae Lim Kook, May 29 2016
a(7)-a(8) corrected by Christopher Cormier, Dec 17 2019
a(9)-a(15) from Christopher Cormier, Dec 19 2019

A329871 Number of static n X n placements of water source-blocks in Minecraft.

Original entry on oeis.org

1, 2, 10, 55, 754, 18853, 82931, 70143802, 11087020614, 3243227117597, 1772826333285009, 1806938280429412270, 3430002591378184399879, 12137184871791092506807847, 80047171080361800628780500638, 983838070049011459232146327319193

Offset: 0

Gus Wiseman, Nov 26 2019

nonn

In Minecraft worlds, a source block of water can be reacted with another source block, two blocks away, linearly or diagonally. This reaction creates a third "infinite" source block in the unoccupied intermediate block or blocks, so called because if the intermediate water source is destroyed or picked up by a player using a bucket, it will immediately regenerate itself.

A placement of water at several positions in an n X n board is said to be static if no infinite water sources are created that are not already present. In particular, the total quantity of water in the system is held constant.

EthosLab, Minecraft - Tutorial: Water
Gus Wiseman, The a(3) = 55 static placements of water source-blocks (black = water).
Christopher Cormier, C# Program

Dominates A273461.
The one-dimensional case is A005251.
Cf. A002416, A006498, A027624, A114901.

vdist[v_,w_]:=Total[Abs[v-w]];
flowdown[prs_]:=Union[prs,With[{ovs=Select[Subsets[prs,{2}],vdist@@#==2&]},Union@@Function[{v,w},Select[Tuples[{Range[Min@@Union[First/@prs],Max@@Union[First/@prs]],Range[Min@@Union[Last/@prs],Max@@Union[Last/@prs]]}],vdist[v,#]==1&&vdist[w,#]==1&]]@@@ovs]];
Table[Length[Select[Subsets[Tuples[Range[n],2]],flowdown[#]==#&]],{n,0,3}]

a(5)-a(6) from Christopher Cormier, Dec 10 2019

a(7)-a(15) from Christopher Cormier, Dec 19 2019

A353429 Number of integer compositions of n with all prime parts and all prime run-lengths.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 4, 0, 2, 2, 5, 4, 9, 1, 5, 12, 20, 11, 19, 18, 31, 43, 54, 37, 63, 95, 121, 124, 154, 178, 261, 353, 393, 417, 565, 770, 952, 1138, 1326, 1647, 2186, 2824, 3261, 3917, 4941, 6423, 7935, 9719, 11554, 14557, 18536, 23380, 27985

Offset: 0

Gus Wiseman, May 16 2022

nonn

			The a(13) = 2 through a(16) = 9 compositions:
  (22333)  (77)       (555)     (3355)
  (33322)  (2255)     (33333)   (5533)
           (5522)     (222333)  (22255)
           (223322)   (333222)  (55222)
           (2222222)            (332233)
                                (2222233)
                                (2223322)
                                (2233222)
                                (3322222)

Alois P. Heinz, Table of n, a(n) for n = 0..4000

The first condition only is A023360, partitions A000607.
For partitions we have A351982, only run-lens A100405, only parts A008483.
The second condition only is A353401, partitions A055923.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A052284 counts compositions into nonprimes, partitions A002095.
A106356 counts compositions by number of adjacent equal parts.
A114901 counts compositions with no runs of length 1, ranked by A353427.
A329738 counts uniform compositions, partitions A047966.
Cf. A005811, A128695, A137200, A167606, A333755, A335464, A353428.

b:= proc(n, h) option remember; `if`(n=0, 1, add(`if`(i<>h and isprime(i),
      add(`if`(isprime(j), b(n-i*j, i), 0), j=2..n/i), 0), i=2..n/2))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..70);  # Alois P. Heinz, May 18 2022

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@PrimeQ/@#&&And@@PrimeQ/@Length/@Split[#]&]],{n,0,15}]

a(26)-a(56) from Alois P. Heinz, May 18 2022

cp's OEIS Frontend

A353430 Number of integer compositions of n that are empty, a singleton, or whose own run-lengths are a consecutive subsequence that is already counted.

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A373952 Number of integer compositions of n whose run-compression sums to 3.

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A329864 Number of compositions of n with the same runs-resistance as cuts-resistance.

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A353427 Numbers k such that the k-th composition in standard order has all run-lengths > 1.

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A353428 Number of integer compositions of n with all parts and all run-lengths > 2.

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A353502 Numbers with all prime indices and exponents > 2.

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A373950 Number of integer compositions of n containing two adjacent ones and no other runs.

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A273461 Number of physically stable n X n placements of water source-blocks in Minecraft.

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