A238125 -id:A238125 - cp's OEIS frontend

A237770 Number of standard Young tableaux with n cells without a succession v, v+1 in a row.

1, 1, 1, 2, 4, 9, 22, 59, 170, 516, 1658, 5583, 19683, 72162, 274796, 1082439, 4406706, 18484332, 79818616, 353995743, 1611041726, 7510754022, 35842380314, 174850257639, 871343536591, 4430997592209, 22978251206350, 121410382810005, 653225968918521

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 13 2014

nonn

A standard Young tableau (SYT) without a succession v, v+1 in a row is called a nonconsecutive tableau.
Also the number of ballot sequences without two consecutive elements equal. A ballot sequence B is a string such that, for all prefixes P of B, h(i)>=h(j) for iA000085).
First column (k=0) of A238125.

			The a(5) = 9 such tableaux of 5 are:
[1]   [2]  [3]   [4]  [5]  [6]  [7]  [8]  [9]
135   13   135   13   13   14   14   15   1
24    24   2     25   2    25   2    2    2
      5    4     4    4    3    3    3    3
                      5         5    4    4
                                          5
The corresponding ballot sequences are:
1:  [ 0 1 0 1 0 ]
2:  [ 0 1 0 1 2 ]
3:  [ 0 1 0 2 0 ]
4:  [ 0 1 0 2 1 ]
5:  [ 0 1 0 2 3 ]
6:  [ 0 1 2 0 1 ]
7:  [ 0 1 2 0 3 ]
8:  [ 0 1 2 3 0 ]
9:  [ 0 1 2 3 4 ]

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..68 (terms 0..48 from Alois P. Heinz)
Timothy Y. Chow, Henrik Eriksson and C. Kenneth Fan, Chess Tableaux, The Electronic Journal of Combinatorics, vol.11, no.2, (2005).
S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.
Wikipedia, Young tableau

Cf. A000085 (all Young tableaux), A000957, A001181, A214021, A214087, A214159, A214875.
Cf. A238126 (tableaux with one succession), A238127 (two successions).
Cf. A264051, A264078.

h:= proc(l, j) option remember; `if`(l=[], 1,
      `if`(l[1]=0, h(subsop(1=[][], l), j-1), add(
      `if`(i<>j and l[i]>0 and (i=1 or l[i]>l[i-1]),
       h(subsop(i=l[i]-1, l), i), 0), i=1..nops(l))))
    end:
g:= proc(n, i, l) `if`(n=0 or i=1, h([1$n, l[]], 0),
      `if`(i<1, 0, g(n, i-1, l)+
      `if`(i>n, 0, g(n-i, i, [i, l[]]))))
    end:
a:= n-> g(n, n, []):
seq(a(n), n=0..30);
# second Maple program (counting ballot sequences):
b:= proc(n, v, l) option remember;
      `if`(n<1, 1, add(`if`(i<>v and (i=1 or l[i-1]>l[i]),
       b(n-1, i, subsop(i=l[i]+1, l)), 0), i=1..nops(l))+
       b(n-1, nops(l)+1, [l[], 1]))
    end:
a:= proc(n) option remember; forget(b); b(n-1, 1, [1]) end:
seq(a(n), n=0..30);

b[n_, v_, l_List] := b[n, v, l] = If[n<1, 1, Sum[If[i != v && (i == 1 || l[[i-1]] > l[[i]]), b[n-1, i, ReplacePart[l, i -> l[[i]]+1]], 0], {i, 1, Length[l]}] + b[n-1, Length[l]+1, Append[l, 1]]]; a[n_] := a[n] = b[n-1, 1, {1}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 06 2015, translated from 2nd Maple program *)

a(n) = Sum_{k=1..A264078(n)} k * A264051(n,k). - Alois P. Heinz, Nov 02 2015

A238126 Number of standard Young tableaux with n cells and exactly one succession.

Original entry on oeis.org

0, 0, 1, 1, 3, 8, 24, 70, 224, 744, 2588, 9317, 34924, 135297, 542123, 2236834, 9508297, 41511215, 186109781, 854874944, 4021672983, 19344343843, 95093249014, 477137036748, 2442413708120, 12742038926613, 67714763161526, 366266085720565, 2015454903261855

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 21 2014

nonn

A succession is a pair of cells (v, v+1) lying in the same row.
Also number of ballot sequences having one flat step.
Second column (k=1) of A238125.

Cf. A237770 (no successions), A238127 (two successions).

A238127 Number of standard Young tableaux with n cells and exactly two successions.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 17, 57, 191, 663, 2415, 9108, 35695, 143989, 599802, 2566917, 11298164, 50967216, 235745644, 1115324000, 5397332497, 26669487517, 134528555379, 691856601631, 3626390958551, 19353306241764, 105122093620388, 580689432523534, 3260906342453966

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 21 2014

nonn

A succession is a pair of cells (v, v+1) lying in the same row.
Also number of ballot sequences having two flat steps.
Third column (k=2) of A238125.

Cf. A237770 (no successions), A238126 (one succession).

A241774 Number of standard Young tableaux with n cells and exactly three successions.

Original entry on oeis.org

1, 2, 9, 29, 108, 399, 1573, 6249, 25642, 107751, 466898, 2071183, 9429453, 43889227, 209111720, 1017353435, 5056450994, 25627110560, 132451351403, 697165582228, 3736595119730, 20372395392794, 112961465325799, 636497947945382, 3643481367551384

Offset: 4

Joerg Arndt and Alois P. Heinz, Apr 28 2014

nonn

A succession is a pair of cells (v, v+1) lying in the same row.

Also number of ballot sequences having three flat steps.

Column k=3 of A238125.

A241775 Number of standard Young tableaux with n cells and exactly four successions.

Original entry on oeis.org

1, 3, 13, 49, 201, 802, 3343, 14368, 62909, 282989, 1299074, 6111610, 29330177, 143869556, 719339872, 3669604633, 19062972671, 100880716332, 543086732800, 2974442015323, 16556687511535, 93655725178808, 537951557765438, 3137040309057826, 18560431188723607

Offset: 5

Joerg Arndt and Alois P. Heinz, Apr 28 2014

nonn

A succession is a pair of cells (v, v+1) lying in the same row.

Also number of ballot sequences having four flat steps.

Column k=4 of A238125.

A241776 Number of standard Young tableaux with n cells and exactly five successions.

Original entry on oeis.org

1, 3, 17, 69, 322, 1408, 6440, 29555, 139005, 662522, 3228150, 15994167, 80849918, 415684406, 2177223330, 11594426886, 62826444538, 345909306526, 1935762918301, 10999013559517, 63460591709283, 371497097807233, 2206388300667674, 13286470507994738

Offset: 6

Joerg Arndt and Alois P. Heinz, Apr 28 2014

nonn

A succession is a pair of cells (v, v+1) lying in the same row.

Also number of ballot sequences having five flat steps.

Column k=5 of A238125.

A241777 Number of standard Young tableaux with n cells and exactly six successions.

Original entry on oeis.org

1, 4, 23, 104, 510, 2411, 11661, 57226, 283833, 1434373, 7346663, 38317226, 202773341, 1091360787, 5961577175, 33093602158, 186413366123, 1066200757106, 6185213644077, 36404110664402, 217206981954701, 1313899575466329, 8052745347023754, 50003529679287381

Offset: 7

Joerg Arndt and Alois P. Heinz, Apr 28 2014

nonn

A succession is a pair of cells (v, v+1) lying in the same row.

Also number of ballot sequences having six flat steps.

Column k=6 of A238125.

A241778 Number of standard Young tableaux with n cells and exactly seven successions.

Original entry on oeis.org

1, 4, 28, 137, 751, 3797, 19868, 103169, 544446, 2892654, 15609326, 85185225, 471914661, 2647616529, 15072126865, 86928421109, 508477806633, 3013152898489, 18098553323837, 110098999926894, 678500886312946, 4233264326440478, 26741870913391272

Offset: 8

Joerg Arndt and Alois P. Heinz, Apr 28 2014

nonn

A succession is a pair of cells (v, v+1) lying in the same row.

Also number of ballot sequences having seven flat steps.

Column k=7 of A238125.

A241779 Number of standard Young tableaux with n cells and exactly eight successions.

Original entry on oeis.org

1, 5, 35, 189, 1088, 5896, 32419, 178832, 989799, 5540426, 31252470, 178571351, 1030696863, 6025834797, 35625675885, 213326126842, 1292327704580, 7927146783432, 49194730886734, 309011198106985, 1963433004838204, 12622320085676279, 82061043600221160

Offset: 9

Joerg Arndt and Alois P. Heinz, Apr 28 2014

nonn

A succession is a pair of cells (v, v+1) lying in the same row.

Also number of ballot sequences having eight flat steps.

Column k=8 of A238125.

A241780 Number of standard Young tableaux with n cells and exactly nine successions.

Original entry on oeis.org

1, 5, 42, 238, 1499, 8635, 50873, 294903, 1724689, 10094399, 59679199, 355198229, 2136972743, 12972026795, 79626155414, 493684963581, 3095482758334, 19612796554079, 125656314294772, 813576800235695, 5325370849930701, 35223931484275717, 235474448496780769

Offset: 10

Joerg Arndt and Alois P. Heinz, Apr 28 2014

nonn

A succession is a pair of cells (v, v+1) lying in the same row.

Also number of ballot sequences having nine flat steps.

Column k=9 of A238125.

cp's OEIS Frontend

A237770 Number of standard Young tableaux with n cells without a succession v, v+1 in a row.

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A238126 Number of standard Young tableaux with n cells and exactly one succession.

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A238127 Number of standard Young tableaux with n cells and exactly two successions.

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A241774 Number of standard Young tableaux with n cells and exactly three successions.

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A241775 Number of standard Young tableaux with n cells and exactly four successions.

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A241776 Number of standard Young tableaux with n cells and exactly five successions.

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A241777 Number of standard Young tableaux with n cells and exactly six successions.

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A241778 Number of standard Young tableaux with n cells and exactly seven successions.

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A241779 Number of standard Young tableaux with n cells and exactly eight successions.

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A241780 Number of standard Young tableaux with n cells and exactly nine successions.

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