A246534 -id:A246534 - cp's OEIS frontend

A104462 Convert the binary strings in A101305 to decimal.

0, 2, 20, 328, 10512, 672800, 86118464, 22046326912, 11287719379200, 11558624644301312, 23672063271529088000, 96960771160183144450048, 794302637344220319334797312, 13013854410247705711981319168000, 426437981314996820770203866497040384

Offset: 0

Jorge Coveiro, Apr 23 2005

The a(n)-th composition in standard order is (2,3,..,n+1), where 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. Moreover, the binary indices of a(n) are row n of A193973. Including 1 gives A164894, reverse A246534. - Gus Wiseman, Jun 28 2022

			From _Gus Wiseman_, Jun 28 2022: (Start)
The terms together with their standard compositions begin:
      0: ()
      2: (2)
     20: (2,3)
    328: (2,3,4)
  10512: (2,3,4,5)
(End)

Michael S. Branicky, Table of n, a(n) for n = 0..80

Cf. A101305.
A version for prime indices is A070826.
Cf. A000120, A002110, A029931, A066099, A070939, A164894, A193973, A233564, A246534, A272919, A333218, A333255.

convert(10,decimal,binary); convert(10100,decimal,binary); convert(101001000,decimal,binary); convert(10100100010000,decimal,binary); convert(10100100010000100000,decimal,binary);

stcinv[q_]:=Total[2^Accumulate[Reverse[q]]]/2;
Table[stcinv[Range[2,n]],{n,8}] (* Gus Wiseman, Jun 28 2022 *)

def a(n): return 0 if n==0 else int("".join("1"+"0"*(i+1) for i in range(n)), 2)
print([a(n) for n in range(15)]) # Michael S. Branicky, Jun 28 2022

a(14) and beyond from Michael S. Branicky, Jun 28 2022

A246533 List of fixed polyominoes in binary coding, ordered by number of bits, then value of the binary code. Can be read as irregular table with row lengths A001168.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 19, 21, 22, 37, 15, 23, 27, 30, 39, 53, 54, 75, 139, 147, 149, 150, 156, 275, 277, 278, 293, 306, 549, 31, 47, 55, 62, 79, 91, 94, 143, 151, 155, 157, 158, 181, 182, 188, 203, 220, 279, 283, 286, 295, 307, 309, 310, 314, 403, 405, 406, 412, 434, 440

Offset: 1

M. F. Hasler, Aug 28 2014

nonn

The binary coding (as suggested in a post to the SeqFan list by F. T. Adams-Watters) is obtained by summing the powers of 2 corresponding to the numbers covered by the polyomino, when the points of the quarter-plane are numbered by antidiagonals, and the animal is pushed to both borders as to obtain the smallest possible value. See example for further details.

The smallest value for an n-omino is the sum 2^0+...+2^(n-1) = 2^n-1 = A000225(n), and the largest value, obtained for the straight n-omino in y direction, is 2^0+2^2+2^5+...+2^(A000217(n)-1) = A246534(n).

			Number the points of the first quadrant as follows:
...
9 ...
5 8 ...
2 4 7 ...
0 1 3 6 10 ...
The "empty" 0-omino is represented by the empty sum equal to 0 = a(1).
The monomino is represented by a square on 0, and the binary code 2^0 = 1 = a(2).
The two fixed dominos are ".." and ":", represented by 2^0+2^1 = 3 = a(3) and 2^0+2^2 = 5 = a(4).
The A001168(3) = 6 fixed trominoes are represented by 2^0+2^1+2^3 = 11 (...), 2^0+2^1+2^2 = 7 (:.), 2^0+2^1+2^4 =19 (.:), ..., 2^0+2^2+2^5 = 37; again these 6 values are listed in increasing size as a(5), ..., a(10).

John Mason, Table of n, a(n) for n = 1..50149
F. T. Adams-Watters, Re: Sequence proposal by John Mason, SeqFan list, Aug 24 2014

See A246521 and A246559 for enumeration of free and one-sided polyominoes.

grow(L,N=[],D=[[1,0],[0,1],[-1,0],[0,-1]])={ for(i=1,#L,for(j=1,#P=L[i],for(k=1,#P,for(d=1,#D, vecmin(P[k]+D[d])<0 && P-=vector(#P,i,D[d])/*shift if needed*/; !setsearch(P,P[k]+D[d]) && N=setunion([setunion([P[k]+D[d]],P)],N); P!=L[i] && P+=vector(#P,i,D[d])/*undo...*/))));if(N,N,[[[0,0]]])}
p2n(P)=sum(i=1,#P,2^(P[i][2]+A000217(P[i][1]+P[i][2])))
for(i=0,5,print(vecsort(apply(p2n,L=if(i,grow(L),[[]])))))

A350250 Numbers k such that the k-th composition in standard order is a non-alternating permutation of an initial interval of positive integers.

Original entry on oeis.org

37, 52, 549, 550, 556, 564, 581, 600, 616, 649, 657, 712, 786, 802, 836, 840, 16933, 16934, 16937, 16940, 16946, 16948, 16965, 16977, 16984, 16994, 17000, 17033, 17041, 17092, 17096, 17170, 17186, 17220, 17224, 17445, 17446, 17452, 17460, 17541, 17569, 17584

Offset: 1

Gus Wiseman, Jan 13 2022

nonn

A sequence is alternating if it is alternately strictly increasing and strictly 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.

			The terms and corresponding permutations begin:
     37: (3,2,1)
     52: (1,2,3)
    549: (4,3,2,1)
    550: (4,3,1,2)
    556: (4,2,1,3)
    564: (4,1,2,3)
    581: (3,4,2,1)
    600: (3,2,1,4)
    616: (3,1,2,4)
    649: (2,4,3,1)
    657: (2,3,4,1)
    712: (2,1,3,4)
    786: (1,4,3,2)
    802: (1,3,4,2)
    836: (1,2,4,3)
    840: (1,2,3,4)
  16933: (5,4,3,2,1)

This is the non-alternating case of A333218.
This is the restriction of A345168 to permutations, complement A345167.
These partitions are counted by A348615, complement A001250.
A003242 counts anti-run compositions, patterns A005649.
A025047 counts alternating compositions, directed A025048/A025049.
A345192 counts non-alternating compositions.
A345194 counts alternating patterns, complement A350252.
Statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of maximal anti-runs is A333381.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Weakly decreasing compositions (partitions) are A114994, strict A333256.
- Weakly increasing compositions (multisets) are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Anti-run compositions are A333489, complement A348612.
Cf. A008965, A059893, A164894, A246534, A333217, A344605, A345162, A350251, A345163, A345171, A345172, A348613.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]==Length[y] &&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[0,1000],(Sort[stc[#]]==Range[Length[stc[#]]]&&!wigQ[stc[#]])&]

A354906 Position of first appearance of n in A354579 = Number of distinct run-lengths of standard compositions.

Original entry on oeis.org

0, 1, 11, 119, 5615, 251871

Offset: 0

Gus Wiseman, Jun 23 2022

			The terms together with their corresponding compositions begin:
       0: ()
       1: (1)
      11: (2,1,1)
     119: (1,1,2,1,1,1)
    5615: (2,2,1,1,1,2,1,1,1,1)
  251871: (1,1,1,2,2,1,1,1,1,2,1,1,1,1,1)

The standard compositions used here are A066099, run-sums A353847/A353932.
The version for partitions is A006939, for run-sums A002110.
For run-sums instead of run-lengths we have A246534 (firsts in A353849).
For runs instead of run-lengths we have A351015 (firsts in A351014).
These are the positions of first appearances in A354579.
A005811 counts runs in binary expansion.
A333627 ranks the run-lengths of standard compositions.
A351596 ranks compositions with distinct run-lengths, counted by A329739.
A353744 ranks compositions with equal run-lengths, counted by A329738.
A353852 ranks compositions with distinct run-sums, counted by A353850.
A353853-A353859 are sequences pertaining to composition run-sum trajectory.
A353860 counts collapsible compositions.
Cf. A003242, A029837, A071625, A124767, A182857, A238279/A333755, A325278, A333381, A333489, A333629.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pd=Table[Length[Union[Length/@Split[stc[n]]]],{n,0,10000}];
Table[Position[pd,n][[1,1]]-1,{n,0,Max@@pd}]

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A104462 Convert the binary strings in A101305 to decimal.

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A246533 List of fixed polyominoes in binary coding, ordered by number of bits, then value of the binary code. Can be read as irregular table with row lengths A001168.

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A350250 Numbers k such that the k-th composition in standard order is a non-alternating permutation of an initial interval of positive integers.

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A354906 Position of first appearance of n in A354579 = Number of distinct run-lengths of standard compositions.

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