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 1-4 of 4 results.

A373949 Triangle read by rows where T(n,k) is the number of integer compositions of n such that replacing each run of repeated parts with a single part (run-compression) yields a composition of k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 1, 2, 4, 0, 1, 0, 4, 4, 7, 0, 1, 1, 5, 6, 5, 14, 0, 1, 0, 6, 10, 10, 14, 23, 0, 1, 1, 6, 14, 12, 29, 26, 39, 0, 1, 0, 9, 16, 19, 40, 54, 46, 71, 0, 1, 1, 8, 22, 22, 64, 82, 96, 92, 124, 0, 1, 0, 10, 26, 30, 82, 137, 144, 204, 176, 214
Offset: 0

Views

Author

Gus Wiseman, Jun 28 2024

Keywords

Examples

			Triangle begins:
   1
   0   1
   0   1   1
   0   1   0   3
   0   1   1   2   4
   0   1   0   4   4   7
   0   1   1   5   6   5  14
   0   1   0   6  10  10  14  23
   0   1   1   6  14  12  29  26  39
   0   1   0   9  16  19  40  54  46  71
   0   1   1   8  22  22  64  82  96  92 124
   0   1   0  10  26  30  82 137 144 204 176 214
   0   1   1  11  32  31 121 186 240 331 393 323 378
Row n = 6 counts the following compositions:
  .  (111111)  (222)  (33)     (3111)   (411)   (6)
                      (2211)   (1113)   (114)   (51)
                      (1122)   (1221)   (1311)  (15)
                      (21111)  (12111)  (1131)  (42)
                      (11112)  (11211)  (2112)  (24)
                               (11121)          (141)
                                                (321)
                                                (312)
                                                (231)
                                                (213)
                                                (132)
                                                (123)
                                                (2121)
                                                (1212)
For example, the composition (1,2,2,1) with compression (1,2,1) is counted under T(6,4).

Links

Crossrefs

Column k = n is A003242 (anti-runs or compressed compositions).
Row-sums are A011782.
Same as A373951 with rows reversed.
Column k = 3 is A373952.
This statistic is represented by A373953, difference A373954.
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.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#]]==k&]], {n,0,10},{k,0,n}]
  • PARI
    T_xy(row_max) = {my(N=row_max+1, x='x+O('x^N), h=1/(1-sum(i=1,N, (y^i*x^i)/(1+x^i*(y^i-1))))); vector(N, n, Vecrev(polcoeff(h, n-1)))}
T_xy(13) \\ John Tyler Rascoe, Mar 20 2025

Formula

G.f.: 1/(1 - Sum_{i>0} (y^i * x^i)/(1 + x^i * (y^i - 1))). - John Tyler Rascoe, Mar 20 2025

A373951 Triangle read by rows where T(n,k) is the number of integer compositions of n such that replacing each run of repeated parts with a single part (run-compression) yields a composition of n - k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 3, 0, 1, 0, 4, 2, 1, 1, 0, 7, 4, 4, 0, 1, 0, 14, 5, 6, 5, 1, 1, 0, 23, 14, 10, 10, 6, 0, 1, 0, 39, 26, 29, 12, 14, 6, 1, 1, 0, 71, 46, 54, 40, 19, 16, 9, 0, 1, 0, 124, 92, 96, 82, 64, 22, 22, 8, 1, 1, 0, 214, 176, 204, 144, 137, 82, 30, 26, 10, 0, 1, 0
Offset: 0

Views

Author

Gus Wiseman, Jun 28 2024

Keywords

Examples

			Triangle begins:
    1
    1   0
    1   1   0
    3   0   1   0
    4   2   1   1   0
    7   4   4   0   1   0
   14   5   6   5   1   1   0
   23  14  10  10   6   0   1   0
   39  26  29  12  14   6   1   1   0
   71  46  54  40  19  16   9   0   1   0
  124  92  96  82  64  22  22   8   1   1   0
Row n = 6 counts the following compositions:
  (6)     (411)   (3111)   (33)     (222)  (111111)  .
  (51)    (114)   (1113)   (2211)
  (15)    (1311)  (1221)   (1122)
  (42)    (1131)  (12111)  (21111)
  (24)    (2112)  (11211)  (11112)
  (141)           (11121)
  (321)
  (312)
  (231)
  (213)
  (132)
  (123)
  (2121)
  (1212)
For example, the composition (1,2,2,1) with compression (1,2,1) is counted under T(6,2).

Crossrefs

Column k = 0 is A003242 (anti-runs or compressed compositions).
Row-sums are A011782.
Same as A373949 with rows reversed.
Column k = 1 is A373950.
This statistic is represented by A373954, difference A373953.
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, A333382, A333489, A373952.

Programs

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

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

Views

Author

Gus Wiseman, Jun 29 2024

Keywords

Comments

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).

Examples

			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)

Links

Crossrefs

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.

Programs

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

Formula

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

Extensions

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

A373955 Numbers k such that the k-th integer composition in standard order contains two adjacent ones and no other runs.

Original entry on oeis.org

3, 11, 14, 19, 27, 28, 29, 35, 46, 51, 56, 57, 67, 75, 78, 83, 91, 92, 93, 99, 110, 112, 113, 114, 116, 118, 131, 139, 142, 155, 156, 157, 163, 179, 184, 185, 195, 203, 206, 211, 219, 220, 221, 224, 225, 226, 229, 230, 232, 233, 236, 237, 259, 267, 270, 275
Offset: 1

Views

Author

Gus Wiseman, Jun 29 2024

Keywords

Comments

Also numbers k such that the excess compression of the k-th integer composition in standard order is 1.
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.
postn of 1 in

Examples

			The terms and corresponding compositions begin:
    3: (1,1)
   11: (2,1,1)
   14: (1,1,2)
   19: (3,1,1)
   27: (1,2,1,1)
   28: (1,1,3)
   29: (1,1,2,1)
   35: (4,1,1)
   46: (2,1,1,2)
   51: (1,3,1,1)
   56: (1,1,4)
   57: (1,1,3,1)
   67: (5,1,1)
   75: (3,2,1,1)
   78: (3,1,1,2)
   83: (2,3,1,1)
   91: (2,1,2,1,1)
   92: (2,1,1,3)
   93: (2,1,1,2,1)
   99: (1,4,1,1)

Links

Crossrefs

These compositions are counted by A373950.
Positions of ones in A373954.
A003242 counts compressed compositions (or anti-runs).
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 encodes compression using compositions in standard order.
A373949 counts compositions by compression-sum.
A373953 gives compression-sum of standard compositions.
Cf. A106356, A124762, A238130, A238279, A238343, A285981, A333382, A333489, A373951, A373952.

Programs

  • Mathematica
    stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[100],Total[stc[#]] == Total[First/@Split[stc[#]]]+1&]
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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