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 12 results. Next

A067228 Number of rectangular standard Young tableaux with n cells.

Original entry on oeis.org

1, 2, 2, 4, 2, 12, 2, 30, 44, 86, 2, 1190, 2, 860, 12014, 26886, 2, 184758, 2, 3359202, 2771342, 117574, 2, 327618902, 701149022, 1485802, 828630662, 27350160662, 2, 808310933492, 2, 2979826568702, 291724349282, 259289582, 557214344578322, 2031957220875002, 2
Offset: 1

Views

Author

Naohiro Nomoto, Feb 20 2002

Keywords

Comments

Number of ways to arrange the numbers 1, 2, .., n=i*j into an i*j rectangle so that each row and each column is increasing.
a(p) = 2 for prime p. - Alois P. Heinz, Jul 25 2012

Links

Crossrefs

Cf. A000085, A060854, A067231.
Column k=0 of A238707.

Programs

  • Maple
    with(numtheory):
a:= n-> n! * add(mul(k!/(i+k)!, k=0..(n/i)-1), i=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 25 2012
  • Mathematica
    a[n_] := n! * Sum[Product[k!/(i+k)!, {k, 0, n/i-1}], {i, Divisors[n]}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jul 02 2015, after Alois P. Heinz *)

Formula

a(n) = n! * Sum_{i|n} Product_{k=0..n/i-1} k!/(i+k)!. - Alois P. Heinz, Jul 25 2012

Extensions

Better name from Joerg Arndt, Feb 24 2014

A244295 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 1.

Original entry on oeis.org

2, 3, 14, 14, 69, 97, 251, 671, 1847, 2111, 12869, 33461, 58343, 189045, 841125, 2207347, 6651215, 12781755, 73096191, 308508927, 904926489, 1727792245, 7638794959, 44017642189, 177969495449, 522668483639, 1662245807549, 4496811662189, 32142974215379
Offset: 3

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 25 2014

Keywords

Comments

Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 1.

Examples

			a(4) = 3:
[1 2]   [1 3]   [1 4]
[3]     [2]     [2]
[4]     [4]     [3]

Links

Crossrefs

Column k=1 of A238707.

Programs

  • Maple
    h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) local j; `if`(n=0 or i<1, 0, `if`(l<>[] and
      l[1]-i=1, `if`(irem(n, i, 'j')=0, h([l[], i$j]), 0),
      add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> g(n, n, []):
seq(a(n), n=3..35);
  • Mathematica
    h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1+l[[i]]-j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, l[[i]]}], {i, n}]];
g[n_, i_, l_] := Module[{j}, If[n == 0 || i<1, 0, If[l != {} && l[[1]]-i == 1, If[j = Quotient[n, i]; Mod[n, i] == 0, h[Join[l, Table[i, {j}]]], 0], Sum[g[n-i*j, i-1, Join[l, Table[i, {j}]]], {j, 0, n/i}]]]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 3, 35}] (* Jean-François Alcover, Aug 25 2021, after Maple code *)

A244296 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 2.

Original entry on oeis.org

3, 6, 35, 71, 295, 751, 2326, 6524, 22309, 55992, 190282, 577410, 1951421, 5414977, 19405654, 64615030, 238446543, 726141375, 2682369977, 9475513873, 41043824531, 138540753071, 524631248766, 1902172512592, 8404692901429, 35025078519164, 148160349275671
Offset: 4

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 25 2014

Keywords

Comments

Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 2.

Links

Crossrefs

Column k=2 of A238707.

Programs

  • Maple
    h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) local j; `if`(n=0 or i<1, 0, `if`(l<>[] and
      l[1]-i=2, `if`(irem(n, i, 'j')=0, h([l[], i$j]), 0),
      add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> g(n, n, []):
seq(a(n), n=4..35);
  • Mathematica
    h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j +
     Sum[If[l[[k]] >= j, 1, 0], {k, i + 1, n}], {j, l[[i]]}], {i, n}]];
g[n_, i_, l_] := Module[{j}, If[n == 0 || i < 1, 0, If[l != {} &&
     l[[1]] - i == 2, If[j = Quotient[n, i]; Mod[n, i] == 0,
     h[Join[l, Table[i, {j}]]], 0], Sum[g[n - i*j, i - 1,
     Join [l, Table[i, {j}]]], {j, 0, n/i}]]]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 4, 35}] (* Jean-François Alcover, Aug 28 2021, after Maple code *)

A244297 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 3.

Original entry on oeis.org

4, 10, 69, 195, 929, 3044, 11824, 40985, 158079, 539876, 2065087, 7272937, 27923757, 101194930, 381940222, 1429135919, 5607176733, 21323561733, 84260636527, 325309822037, 1337034045619, 5421586411034, 22509005469068, 92412147570641, 390023528935516
Offset: 5

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 25 2014

Keywords

Comments

Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 3.

Links

Crossrefs

Column k=3 of A238707.

Programs

  • Maple
    h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
    add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
g:= proc(n, i, l) local j; `if`(n=0 or i<1, 0, `if`(l<>[] and
      l[1]-i=3, `if`(irem(n, i, 'j')=0, h([l[], i$j]), 0),
      add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> g(n$2, []):
seq(a(n), n=5..35);
  • Mathematica
    h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i + 1, n}], {j, [[i]]}], {i, n}]];
g[n_, i_, l_] := Module[{j}, If[n == 0 || i < 1, 0, If[l != {} && l[[1]] - i == 3, If[j = Quotient[n, i]; Mod[n, i] == 0, h[Join[l, Table[i, {j}]]], 0], Sum[g[n - i*j, i - 1, Join[l, Table[i, {j}]]], {j, 0, n/i}]]]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 5, 35}] (* Jean-François Alcover, Aug 28 2021, after Maple code *)

A244298 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 4.

Original entry on oeis.org

5, 15, 119, 421, 2254, 8999, 40349, 166817, 737829, 3008774, 13186593, 54944783, 238422808, 1010671048, 4395831546, 18821162274, 82799233661, 359711480525, 1599420076729, 7030074945271, 31626819884986, 141486845119777, 646988113794544, 2940338763342920
Offset: 6

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 25 2014

Keywords

Comments

Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 4.

Links

Crossrefs

Column k=4 of A238707.

Programs

  • Maple
    h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
    add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
g:= proc(n, i, l) local j; `if`(n=0 or i<1, 0, `if`(l<>[] and
      l[1]-i=4, `if`(irem(n, i, 'j')=0, h([l[], i$j]), 0),
      add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> g(n$2, []):
seq(a(n), n=6..35);
  • Mathematica
    h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i + 1, n}], {j, l[[i]]}], {i, n}]];
g[n_, i_, l_] := Module[{j}, If[n == 0 || i < 1, 0, If[l != {} && l[[1]] - i == 4, If[j = Quotient[n, i]; Mod[n, i] == 0, h[Join[l, Table[i, {j}]]], 0], Sum[g[n - i*j, i - 1, Join[l, Table[i, {j}]]], {j, 0, n/i}]]]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 6, 35}] (* Jean-François Alcover, Aug 28 2021, after Maple code *)

A244299 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 5.

Original entry on oeis.org

6, 21, 188, 791, 4696, 21614, 109745, 513421, 2557358, 11885545, 58291639, 275421640, 1342532532, 6411950652, 31310737486, 151220406569, 742729520457, 3625802212441, 17956348335989, 88575381634494, 442565032597013, 2207206278880826, 11138577085071310
Offset: 7

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 25 2014

Keywords

Comments

Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 5.

Links

Crossrefs

Column k=5 of A238707.

Programs

  • Maple
    h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
    add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
g:= proc(n, i, l) local j; `if`(n=0 or i<1, 0, `if`(l<>[] and
      l[1]-i=5, `if`(irem(n, i, 'j')=0, h([l[], i$j]), 0),
      add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> g(n$2, []):
seq(a(n), n=7..35);
  • Mathematica
    h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i + 1, n}], {j, l[[i]]}], {i, n}]];
g[n_, i_, l_] := Module[{j}, If[n == 0 || i < 1, 0, If[l != {} && l[[1]] - i == 5, If[j = Quotient[n, i]; Mod[n, i] == 0, h[Join[l, Table[i, {j}]]], 0], Sum[g[n - i*j, i - 1, Join[l, Table[i, {j}]]], {j, 0, n/i}]]]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 7, 35}] (* Jean-François Alcover, Aug 28 2021, after Maple code *)

A244300 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 6.

Original entry on oeis.org

7, 28, 279, 1354, 8823, 45553, 256192, 1328368, 7272035, 37498159, 201732490, 1052038304, 5628260010, 29642509180, 158744001098, 844461334762, 4549886593291, 24435491901926, 132677029062176, 719558882421952, 3940213225673584, 21584248413514700
Offset: 8

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 25 2014

Keywords

Comments

Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 6.

Links

Crossrefs

Column k=6 of A238707.

Programs

  • Maple
    h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
    add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
g:= proc(n, i, l) local j; `if`(n=0 or i<1, 0, `if`(l<>[] and
      l[1]-i=6, `if`(irem(n, i, 'j')=0, h([l[], i$j]), 0),
      add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> g(n$2, []):
seq(a(n), n=8..35);

A244301 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 7.

Original entry on oeis.org

8, 36, 395, 2166, 15365, 87515, 537860, 3043387, 18079805, 101484440, 590719670, 3340021038, 19305133044, 110029379692, 636377579038, 3658493392345, 21277586564634, 123462793881974, 723417146721998, 4238255849152496, 25044116796884830, 148185594856438664
Offset: 9

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 25 2014

Keywords

Comments

Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 7.

Links

Crossrefs

Column k=7 of A238707.

Programs

  • Maple
    h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
    add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
g:= proc(n, i, l) local j; `if`(n=0 or i<1, 0, `if`(l<>[] and
      l[1]-i=7, `if`(irem(n, i, 'j')=0, h([l[], i$j]), 0),
      add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> g(n$2, []):
seq(a(n), n=9..35);

A244302 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 8.

Original entry on oeis.org

9, 45, 539, 3290, 25234, 156743, 1042823, 6374389, 40710734, 245996972, 1533537330, 9295148728, 57412881670, 349869872571, 2159393201713, 13252915145611, 82161769477646, 508497521855467, 3174435344149894, 19827435510586970, 124802677329672826
Offset: 10

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 25 2014

Keywords

Comments

Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 8.

Links

Crossrefs

Column k=8 of A238707.

Programs

  • Maple
    h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
    add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
g:= proc(n, i, l) local j; `if`(n=0 or i<1, 0, `if`(l<>[] and
      l[1]-i=8, `if`(irem(n, i, 'j')=0, h([l[], i$j]), 0),
      add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> g(n$2, []):
seq(a(n), n=10..35);

A244303 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 9.

Original entry on oeis.org

10, 55, 714, 4796, 39544, 265589, 1899137, 12448912, 84901024, 547968340, 3633493460, 23423908430, 153474667719, 991845819899, 6480618983179, 42093506667304, 275840531014103, 1804204772698796, 11893232452570720, 78437868094585319, 521001980260102004
Offset: 11

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 25 2014

Keywords

Comments

Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 9.

Links

Crossrefs

Column k=9 of A238707.

Programs

  • Maple
    h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
    add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
g:= proc(n, i, l) local j; `if`(n=0 or i<1, 0, `if`(l<>[] and
      l[1]-i=9, `if`(irem(n, i, 'j')=0, h([l[], i$j]), 0),
      add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> g(n$2, []):
seq(a(n), n=11..35);
Showing 1-10 of 12 results. Next

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