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 1-10 of 14 results. Next

A299968 Number of normal generalized Young tableaux of size n with all rows and columns strictly increasing.

Original entry on oeis.org

1, 1, 2, 5, 15, 51, 189, 753, 3248, 14738, 70658, 354178, 1857703, 10121033, 57224955, 334321008, 2017234773, 12530668585, 80083779383, 525284893144, 3533663143981, 24336720018666, 171484380988738, 1234596183001927, 9075879776056533, 68052896425955296
Offset: 0

Views

Author

Gus Wiseman, Feb 26 2018

Keywords

Comments

A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers.

Examples

			The a(4) = 15 tableaux:
1 2 3 4
.
1 2 3   1 2 4   1 3 4   1 2 3   1 2 3
4       3       2       2       3
.
1 2   1 3   1 2
3 4   2 4   2 3
.
1 2   1 3   1 2   1 4   1 3
3     2     2     2     2
4     4     3     3     3
.
1
2
3
4

Links

Crossrefs

Cf. A000085, A000898, A001221, A005117, A006958, A015128, A138178, A238121, A238690, A285175, A296561, A297388, A299926, A300120, A300122.

Programs

  • Mathematica
    unddis[y_]:=DeleteCases[y-#,0]&/@Tuples[Table[If[y[[i]]>Append[y,0][[i+1]],{0,1},{0}],{i,Length[y]}]];
dos[y_]:=With[{sam=Rest[unddis[y]]},If[Length[sam]===0,If[Total[y]===0,{{}},{}],Join@@Table[Prepend[#,y]&/@dos[sam[[k]]],{k,1,Length[sam]}]]];
Table[Sum[Length[dos[y]],{y,IntegerPartitions[n]}],{n,1,8}]

Formula

a(n) = Sum_{k=0..n} 2^k * A238121(n,k). - Ludovic Schwob, Sep 23 2023

Extensions

More terms from Ludovic Schwob, Sep 23 2023

A238122 Irregular triangle read by rows: T(n,k) gives the number of ballot sequences of length n having k descents, n>=0, 0<=k<=A083920(n-1).

Original entry on oeis.org

1, 1, 2, 3, 1, 5, 5, 7, 16, 3, 11, 43, 21, 1, 15, 99, 101, 17, 22, 215, 373, 145, 9, 30, 430, 1174, 836, 146, 4, 42, 834, 3337, 3846, 1324, 112, 1, 56, 1529, 8642, 15002, 8786, 1615, 66, 77, 2765, 21148, 52132, 47013, 15403, 1582, 32, 101, 4792, 48713, 164576, 214997, 112106, 21895, 1310, 14
Offset: 0

Views

Author

Joerg Arndt and Alois P. Heinz, Feb 21 2014

Keywords

Comments

Same as A238121, with zeros omitted.
Columns k=0-10 give: A000041, A241794, A241795, A241796, A241797, A241798, A241799, A241800, A241801, A241802, A241803.
T(2n,n) gives A241804.
T(2n+1,n) gives A241805.
Row sums are A000085.
T(n*(n+1)/2,n*(n-1)/2) = 1.

Examples

			T(5,0) = 7: [1,1,1,1,1], [1,1,1,1,2], [1,1,1,2,2], [1,1,1,2,3], [1,1,2,2,3], [1,1,2,3,4], [1,2,3,4,5].
T(5,1) = 16: [1,1,1,2,1], [1,1,2,1,1], [1,1,2,1,2], [1,1,2,1,3], [1,1,2,2,1], [1,1,2,3,1], [1,1,2,3,2], [1,2,1,1,1], [1,2,1,1,2], [1,2,1,1,3], [1,2,1,2,3], [1,2,1,3,4], [1,2,3,1,1], [1,2,3,1,2], [1,2,3,1,4], [1,2,3,4,1].
T(5,2) = 3: [1,2,1,2,1], [1,2,1,3,1], [1,2,1,3,2].
Triangle starts:
00:   1;
01:   1;
02:   2;
03:   3,    1;
04:   5,    5;
05:   7,   16,      3;
06:  11,   43,     21,      1;
07:  15,   99,    101,     17;
08:  22,  215,    373,    145,      9;
09:  30,  430,   1174,    836,    146,      4;
10:  42,  834,   3337,   3846,   1324,    112,      1;
11:  56, 1529,   8642,  15002,   8786,   1615,     66;
12:  77, 2765,  21148,  52132,  47013,  15403,   1582,    32;
13: 101, 4792,  48713, 164576, 214997, 112106,  21895,  1310,  14;
14: 135, 8216, 108147, 484609, 874413, 672015, 215849, 26159, 932, 5;
...

Links

Programs

  • Maple
    b:= proc(n, v, l) option remember; `if`(n<1, 1, expand(
      add(`if`(i=1 or l[i-1]>l[i], `if`(i(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n-1, 1, [1])):
seq(T(n), n=0..14);
  • Mathematica
    b[n_, v_, l_List] := b[n, v, l] = If[n<1, 1, Expand[Sum[If[i == 1 || l[[i-1]] > l[[i]], If[i l[[i]]+1]], 0], {i, 1, Length[ l ]}] + b[n-1, Length[l]+1, Append[l, 1]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n-1, 1, {1}]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Maple *)

A241794 Number of ballot sequences of length n having exactly one descent.

Original entry on oeis.org

1, 5, 16, 43, 99, 215, 430, 834, 1529, 2765, 4792, 8216, 13684, 22575, 36402, 58285, 91617, 143097, 220317, 337100, 509718, 766655, 1141457, 1690994, 2484138, 3631902, 5271297, 7617750, 10937657, 15640771, 22240250, 31502316, 44396662, 62345539, 87149572
Offset: 3

Views

Author

Joerg Arndt and Alois P. Heinz, Apr 28 2014

Keywords

Examples

			a(3) = 1: [1,2,1].
a(4) = 5: [1,1,2,1], [1,2,1,1], [1,2,1,2], [1,2,1,3], [1,2,3,1].

Crossrefs

Column k=1 of A238121 and of A238122.

A241795 Number of ballot sequences of length n having exactly two descents.

Original entry on oeis.org

3, 21, 101, 373, 1174, 3337, 8642, 21148, 48713, 108147, 229984, 476430, 955667, 1879169, 3603751, 6801106, 12584048, 22974152, 41268291, 73301654, 128441712, 222871187, 382322022, 650209758, 1094918922, 1829777544, 3031447986, 4987910871, 8144753275
Offset: 5

Views

Author

Joerg Arndt and Alois P. Heinz, Apr 28 2014

Keywords

Examples

			a(5) = 3: [1,2,1,2,1], [1,2,1,3,1], [1,2,1,3,2].

Crossrefs

Column k=2 of A238121 and of A238122.

A241796 Number of ballot sequences of length n having exactly three descents.

Original entry on oeis.org

1, 17, 145, 836, 3846, 15002, 52132, 164576, 484609, 1337597, 3521517, 8850458, 21485321, 50358968, 114911640, 255114977, 554333346, 1178070741, 2459840199, 5042845917, 10185374250, 20257833206, 39778823250, 77088115856, 147739793527, 279923493220
Offset: 6

Views

Author

Joerg Arndt and Alois P. Heinz, Apr 28 2014

Keywords

Examples

			a(6) = 1: [1,2,1,3,2,1].

Crossrefs

Column k=3 of A238121 and of A238122.

A241797 Number of ballot sequences of length n having exactly four descents.

Original entry on oeis.org

9, 146, 1324, 8786, 47013, 214997, 874413, 3228005, 11049817, 35389760, 107476721, 310853113, 863984856, 2313084483, 6003120701, 15122836712, 37154716734, 89093938147, 209258012995, 481663102867, 1089445474512, 2422311605452, 5305675093522, 11451261202545
Offset: 8

Views

Author

Joerg Arndt and Alois P. Heinz, Apr 28 2014

Keywords

Examples

			a(8) = 9: [1,2,1,2,1,3,2,1], [1,2,1,3,1,4,2,1], [1,2,1,3,2,1,2,1], [1,2,1,3,2,1,3,1], [1,2,1,3,2,1,3,2], [1,2,1,3,2,1,4,1], [1,2,1,3,2,1,4,2], [1,2,1,3,2,1,4,3], [1,2,1,3,2,4,3,1].

Crossrefs

Column k=4 of A238121 and of A238122.

A241798 Number of ballot sequences of length n having exactly five descents.

Original entry on oeis.org

4, 112, 1615, 15403, 112106, 672015, 3477691, 16037089, 67235292, 260887532, 946435074, 3246205986, 10587385911, 33085654876, 99411959668, 288773319190, 812777159786, 2225529375130, 5937743304924, 15483111029378, 39503358108034, 98849770060367, 242801184657314
Offset: 9

Views

Author

Joerg Arndt and Alois P. Heinz, Apr 28 2014

Keywords

Examples

			a(9) = 4: [1,2,1,3,2,1,3,2,1], [1,2,1,3,2,1,4,2,1], [1,2,1,3,2,1,4,3,1], [1,2,1,3,2,1,4,3,2].

Crossrefs

Column k=5 of A238121 and of A238122.

A241799 Number of ballot sequences of length n having exactly six descents.

Original entry on oeis.org

1, 66, 1582, 21895, 215849, 1685957, 11051141, 63115364, 322683730, 1503863212, 6488505684, 26177296409, 99733640537, 361023920432, 1250117445742, 4157235614630, 13341716405949, 41433896651428, 124970671492315, 366791960890982, 1050487889623924
Offset: 10

Views

Author

Joerg Arndt and Alois P. Heinz, Apr 28 2014

Keywords

Examples

			a(10) = 1: [1,2,1,3,2,1,4,3,2,1].

Crossrefs

Column k=6 of A238121 and of A238122.

A241800 Number of ballot sequences of length n having exactly seven descents.

Original entry on oeis.org

32, 1310, 26159, 349021, 3537427, 29216728, 205795528, 1274811402, 7106683612, 36230838069, 171229815244, 757346853197, 3162754360421, 12546713825786, 47572219652564, 173111049909503, 607264725020735, 2059610656640095, 6776341942347065, 21674530092011291
Offset: 12

Views

Author

Joerg Arndt and Alois P. Heinz, Apr 28 2014

Keywords

Crossrefs

Column k=7 of A238121 and of A238122.

A241801 Number of ballot sequences of length n having exactly eight descents.

Original entry on oeis.org

14, 932, 26865, 486170, 6385041, 66251021, 572940736, 4284412169, 28422470746, 170581304048, 939549814317, 4806572652156, 23042550110214, 104344981470162, 448963898693786, 1845829609158989, 7281093726825326, 27670464821261399, 101613723592257039
Offset: 13

Views

Author

Joerg Arndt and Alois P. Heinz, Apr 28 2014

Keywords

Examples

			a(13) = 14: [1,2,1,3,2,1,3,2,1,4,3,2,1], [1,2,1,3,2,1,4,2,1,5,3,2,1], [1,2,1,3,2,1,4,3,1,5,4,2,1], [1,2,1,3,2,1,4,3,2,1,3,2,1], [1,2,1,3,2,1,4,3,2,1,4,2,1], [1,2,1,3,2,1,4,3,2,1,4,3,1], [1,2,1,3,2,1,4,3,2,1,4,3,2], [1,2,1,3,2,1,4,3,2,1,5,2,1], [1,2,1,3,2,1,4,3,2,1,5,3,1], [1,2,1,3,2,1,4,3,2,1,5,3,2], [1,2,1,3,2,1,4,3,2,1,5,4,1], [1,2,1,3,2,1,4,3,2,1,5,4,2], [1,2,1,3,2,1,4,3,2,1,5,4,3], [1,2,1,3,2,1,4,3,2,5,4,3,1].

Crossrefs

Column k=8 of A238121 and of A238122.
Showing 1-10 of 14 results. Next

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

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