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.

Previous Showing 11-15 of 15 results.

A352514 Number of strong nonexcedances (parts below the diagonal) of the n-th composition in standard order.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 1, 2, 1, 3, 3, 4, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 1, 2, 1, 3, 3, 4, 0, 1, 1, 2, 0, 2, 2, 3, 1, 2, 3, 4, 3, 4, 4, 5, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 1, 2, 1, 3, 3, 4, 0, 1, 1, 2, 0, 2, 2
Offset: 0

Views

Author

Gus Wiseman, Mar 22 2022

Keywords

Comments

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. See also A000120, A059893, A070939, A114994, A225620.

Examples

			The 83rd composition in standard order is (2,3,1,1), with strong nonexcedances {3,4}, so a(83) = 2.

Links

Crossrefs

Positions of first appearances are A000225.
The weak version is A352515, counted by A352522 (first column A238874).
The opposite version is A352516, counted by A352524 (first column A008930).
The weak opposite version is A352517, counted by A352525 (first A177510).
The triangle A352521 counts these compositions (first column A219282).
A008292 is the triangle of Eulerian numbers (version without zeros).
A011782 counts compositions.
A173018 counts permutations by number of excedances, weak A123125.
A238349 counts comps by fixed parts, first col A238351, rank stat A352512.
A352490 is the (strong) nonexcedance set of A122111.
A352523 counts comps by unfixed parts, first col A010054, rank stat A352513.
Cf. A088218, A098825, A114088, A115994, A131577, A238352.

Programs

  • Mathematica
    stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pa[y_]:=Length[Select[Range[Length[y]],#>y[[#]]&]];
Table[pa[stc[n]],{n,0,30}]

A352515 Number of weak nonexcedances (parts on or below the diagonal) of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 1, 2, 1, 3, 3, 4, 0, 1, 1, 2, 0, 2, 2, 3, 1, 2, 3, 4, 3, 4, 4, 5, 0, 1, 1, 2, 0, 2, 2, 3, 0, 1, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 3, 4, 4, 5, 2, 4, 4, 5, 4, 5, 5, 6, 0, 1, 1, 2, 0, 2, 2, 3, 0, 1, 2, 3, 2, 3, 3, 4, 0, 1, 1, 2, 2, 3, 3
Offset: 0

Views

Author

Gus Wiseman, Mar 23 2022

Keywords

Comments

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. See also A000120, A059893, A070939, A114994, A225620.

Examples

			The 89th composition in standard order is (2,1,3,1), with weak nonexcedances {2,3,4}, so a(89) = 3.

Links

Crossrefs

Positions of first appearances are A000225.
The strong version is A352514, counted by A352521 (first column A219282).
The strong opposite version is A352516, counted by A352524 (first A008930).
The opposite version is A352517, counted by A352525 (first column A177510).
Triangle A352522 counts these comps (first col A238874), partitions A115994.
A008292 is the triangle of Eulerian numbers (version without zeros).
A011782 counts compositions.
A173018 counts permutations by number of excedances, weak A123125.
A238349 counts comps by fixed points, first col A238351, rank stat A352512.
A352488 is the weak nonexcedance set of A122111.
A352523 counts comps by unfixed pts, first col A010054, rank stat A352513.
Cf. A088218, A098825, A114088, A131577, A238352.

Programs

  • Mathematica
    stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
paw[y_]:=Length[Select[Range[Length[y]],#>=y[[#]]&]];
Table[paw[stc[n]],{n,0,30}]

A352517 Number of weak excedances (parts on or above the diagonal) of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2
Offset: 0

Views

Author

Gus Wiseman, Mar 23 2022

Keywords

Comments

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. See also A000120, A059893, A070939, A114994, A225620.

Examples

			The 169th composition in standard order is (2,2,3,1), with weak excedances {1,2,3}, so a(169) = 3.

Links

Crossrefs

Positive positions of first appearances are A164894.
The version for partitions is A257990.
The strong opposite version is A352514, counted by A352521 (first A219282).
The opposite version is A352515, counted by A352522 (first column A238874).
The strong version is A352516, counted by A352524 (first column A008930).
The triangle A352525 counts these compositions (first column A177510).
A008292 is the triangle of Eulerian numbers (version without zeros).
A011782 counts compositions.
A173018 counts permutations by number of excedances, weak A123125.
A238349 counts comps by fixed points, first col A238351, rank stat A352512.
A352489 is the weak excedance set of A122111.
A352523 counts comps by unfixed points, first A010054, rank stat A352513.
Cf. A088218, A098825, A114088, A115994, A131577, A238352, A352513, A352523.

Programs

  • Mathematica
    stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pdw[y_]:=Length[Select[Range[Length[y]],#<=y[[#]]&]];
Table[pdw[stc[n]],{n,0,30}]

A351983 Number of integer compositions of n with exactly one part above the diagonal.

Original entry on oeis.org

0, 0, 1, 2, 5, 9, 18, 35, 67, 131, 257, 505, 996, 1973, 3915, 7781, 15486, 30855, 61527, 122764, 245069, 489412, 977673, 1953515, 3904108, 7803545, 15599618, 31187269, 62355347, 124679883, 249310255, 498540890, 996953659, 1993701032, 3987069747, 7973603891
Offset: 0

Views

Author

Gus Wiseman, Apr 02 2022

Keywords

Examples

			The a(2) = 1 through a(6) = 18 compositions:
  (2)  (3)   (4)    (5)     (6)
       (21)  (13)   (14)    (15)
             (22)   (32)    (42)
             (31)   (41)    (51)
             (211)  (131)   (114)
                    (212)   (132)
                    (221)   (141)
                    (311)   (213)
                    (2111)  (222)
                            (312)
                            (321)
                            (411)
                            (1311)
                            (2112)
                            (2121)
                            (2211)
                            (3111)
                            (21111)

Links

Crossrefs

The version for permutations is A000295, weak A057427.
The version for partitions is A002620, weak A001477.
The weak version is A177510.
The version for fixed points is A240736, nonfixed A352520.
This is column k = 1 of A352524; column k = 0 is A008930.
A238349 counts compositions by fixed points, first column A238351.
A352521 counts compositions by strong nonexcedances, first column A219282.
A352522 counts compositions by weak nonexcedances, first column A238874.
A352523 counts compositions by nonfixed points, first column A010054.
A352524 counts compositions by strong excedances, first column A008930.
A352525 counts compositions by weak excedances, first column A177510.
Cf. A088218, A098825, A115994, A238352, A330644, A352516.

Programs

  • Mathematica
    pless[y_]:=Length[Select[Range[Length[y]],#
  • PARI
    S(v,u,c=0)={vector(#v, k, c + sum(i=1, k-1, v[k-i]*u[i]))}
seq(n)={my(v=vector(1+n), s=0); v[1]=1; for(i=1, n, v=S(v, vector(n, j, if(j>i,'x,1)), O(x^2)); s+=apply(p->polcoef(p,1), v)); s} \\ Andrew Howroyd, Jan 02 2023

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 02 2023

A352875 Number of integer compositions y of n with a fixed point y(i) = i.

Original entry on oeis.org

0, 1, 1, 2, 5, 10, 21, 42, 86, 174, 351, 708, 1424, 2861, 5743, 11520, 23092, 46269, 92673, 185562, 371469, 743491, 1487870, 2977164, 5956616, 11916910, 23839736, 47688994, 95393322, 190811346, 381662507, 763389209, 1526881959, 3053930971, 6108131542, 12216698288
Offset: 0

Views

Author

Gus Wiseman, May 15 2022

Keywords

Examples

			The a(0) = 0 through a(5) = 10 compositions (empty column indicated by dot):
  .  (1)  (11)  (12)   (13)    (14)
                (111)  (22)    (32)
                       (112)   (113)
                       (121)   (122)
                       (1111)  (131)
                               (221)
                               (1112)
                               (1121)
                               (1211)
                               (11111)

Links

Crossrefs

The version for partitions is A001522, ranked by A352827 (unproved).
The version for permutations is A002467, complement A000166.
The complement for partitions is A064428, ranked by A352826 (unproved).
This is the sum of latter columns of A238349, nonfixed A352523.
The complement is counted by A238351.
The complement for reversed partitions is A238394, ranked by A352830.
The version for reversed partitions is A238395, ranked by A352872.
The case of just one fixed point is A240736.
A008290 counts permutations by fixed points, nonfixed A098825.
A011782 counts compositions.
A115720 and A115994 count partitions by Durfee square.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A352512 counts fixed points in standard compositions, nonfixed A352513.
A352521 = comps by subdiags, first col A219282, rank stat A352514.
A352522 = comps by weak subdiags, first col A238874, rank stat A352515.
A352524 = comps by superdiags, first col A008930, rank stat A352516.
A352525 = comps by weak superdiags, col k=1 A177510, rank stat A352517.
A352833 counts partitions by fixed points.
Cf. A006918, A008292, A088218, A114088, A123125, A188674, A257990.

Programs

  • Mathematica
    pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pq[#]>0&]],{n,0,15}]
  • PARI
    S(v,u,c)={vector(#v, k, c + sum(i=1, k-1, v[k-i]*u[i]))}
seq(n)={my(v=vector(1+n), s=vector(#v, i, 2^(i-2))); v[1]=1; s[1]=0; for(i=1, n, v=S(v, vector(n, j, if(j==i,'x,1)), O(x)); s-=apply(p->polcoef(p,0), v)); s} \\ Andrew Howroyd, Jan 02 2023

Formula

a(n) = 2^(n-1) - A238351(n) for n >= 1. - Andrew Howroyd, Jan 02 2023

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 02 2023
Previous Showing 11-15 of 15 results.

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