A208615 -id:A208615 - cp's OEIS frontend

A185148 Number of rectangular arrangements of [1,3n] in 3 increasing sequences of size n and n monotonic sequences of size 3.

1, 6, 53, 587, 7572, 109027, 1705249, 28440320, 499208817, 9134237407, 172976239886, 3371587949969, 67351686970929, 1374179898145980, 28557595591148315, 603118526483125869, 12920388129877471030, 280324904918707937001, 6151595155000424589327, 136384555249451824930126

Offset: 1

Olivier Gérard, Feb 15 2011

nonn

a(n) counts a subset of A025035(n).
a(n) counts a more general set than A005789(n).
a(n) is also the number of (3*n-1)-step walks on 3-dimensional cubic lattice from (1,0,0) to (n,n,n) with steps in {(1,0,0), (0,1,0), (0,0,1)} such that for each point (x,y,z) we have x<=y<=z or x>=y>=z. - Alois P. Heinz, Feb 29 2012

			For n = 2 the a(2) = 6 arrangements are:
+---+  +---+  +---+  +---+  +---+  +---+
|1 4|  |1 6|  |1 3|  |1 3|  |1 2|  |1 2|
|2 5|  |2 5|  |2 5|  |2 4|  |3 5|  |3 4|
|3 6|  |3 4|  |4 6|  |5 6|  |4 6|  |5 6|
+---+  +---+  +---+  +---+  +---+  +---+
Only the second of these arrangements is not counted by A005789(2).

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..700 (terms 0..200 from Alois P. Heinz)

Cf. A025035, A005789.

Column k=3 of A208615. - Alois P. Heinz, Feb 29 2012

b:= proc(x, y, z) option remember;
      `if`(x=z, `if`(x=0, 1, 2*b(x-1, y, z)), `if`(x>0, b(x-1, y, z), 0)+
      `if`(y>x, b(x, y-1, z), 0)+ `if`(z>y, b(x, y, z-1), 0))
    end:
a:= n-> b(n-1, n$2):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 29 2012

b[x_, y_, z_] := b[x, y, z] = If[x == z, If[x == 0, 1, 2*b[x - 1, y, z]], If[x > 0, b[x - 1, y, z], 0] + If[y > x, b[x, y - 1, z], 0] + If[z > y, b[x, y, z - 1], 0]];
a[n_] := b[n - 1, n, n];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 12 2017, after Alois P. Heinz *)

a(n) ~ c * 27^n / n^4, where c = 0.608287207375... . - Vaclav Kotesovec, Sep 03 2014, updated Sep 07 2016

More terms and example from Alois P. Heinz, Feb 22 2011

Extended beyond a(8) by Alois P. Heinz, Feb 22 2012

A208616 Number of Young tableaux with 3 n-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).

Original entry on oeis.org

1, 1, 10, 53, 491, 6091, 87781, 1386529, 23374495, 414325055, 7646034683, 145862292213, 2861143072425, 57468095412921, 1178095930854841, 24584089994286121, 521086299342539671, 11198784502153759831, 243661974373753909051, 5360563436205104422681

Offset: 0

Alois P. Heinz, Feb 29 2012

Also the number of (3*n-1)-step walks on n-dimensional cubic lattice from (1,0,...,0) to (3,3,...,3) with positive unit steps in all dimensions such that for each point (p_1,p_2,...,p_n) we have p_1<=p_2<=...<=p_n or p_1>=p_2>=...>=p_n.

Alois P. Heinz, Table of n, a(n) for n = 0..706

Row n=3 of A208615.

a:= proc(n) option remember; `if`(n<5, [1, 1, 10, 53, 491][n+1],
      ((116013096898*n^6 -1106227006064*n^5 +3651730072724*n^4
      -5019246600372*n^3 +2923780805838*n^2 -701199942904*n) *a(n-1)
      +(-429126244301*n^6 +4283495440027*n^5 -14793057372915*n^4
      +19089754215809*n^3 -168467698444*n^2 -17547244920336*n
      +9564646580160) *a(n-2) +(24700698282*n^6 +2323122442728*n^5
      -31157649402714*n^4 +153639646198428*n^3 -363480023453028*n^2
      +415894667210784*n -184360926114960) *a(n-3) +(292122384552*n^6
      -5522876986500*n^5 +42303228071580*n^4 -167574646102140*n^3
      +360649174254588*n^2 -397826818736400*n +174796279534800) *a(n-4))/
      (n*(3709935431*n^5 -22486109809*n^4 +4251368675*n^3 +135507711725*n^2
      -75536091046*n -180596388856)))
    end:
seq(a(n), n=0..30);

b[nn__] := b[nn] = If[(lg = Length[{nn}]) < 2, 1, If[First[{nn}] == Last[{nn}], If[First[{nn}] == 0, 1, 2*b[First[{nn}]-1, Sequence @@ Rest[{nn}]]], If[First[{nn}] > 0, b[First[{nn}] - 1, Sequence @@ Rest[{nn}]], 0] + Sum[If[{nn}[[j]] > {nn}[[j-1]], b[Sequence @@ Table[ {nn}[[i]] - If[i == j, 1, 0], {i, 1, lg}]], 0], {j, 2, lg}]]];
a[n_] := If[n == 0, 1, b[2, Sequence @@ Table[3, {n-1}]]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 29 2017, after Alois P. Heinz (cf. A208615) *)

a(n) ~ 3^(3*n+1/2) / (Pi*n^4). - Vaclav Kotesovec, Jul 16 2014

A208624 Number of Young tableaux with n 4-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).

Original entry on oeis.org

1, 1, 15, 491, 25187, 1725819, 144558247, 14029729645, 1523926182363, 180929760551225, 23086562828397479, 3126799551978895629, 445266632168280620515, 66178991463387525289801, 10206120232877820185701707, 1625518539321873371313790283

Offset: 0

Alois P. Heinz, Feb 29 2012

Also the number of (4*n-1)-step walks on 4-dimensional cubic lattice from (1,0,0,0) to (n,n,n,n) with positive unit steps in all dimensions such that for each point (p_1,p_2,p_3,p_4) we have p_1<=p_2<=p_3<=p_4 or p_1>=p_2>=p_3>=p_4.

Alois P. Heinz, Table of n, a(n) for n = 0..100

Column k=4 of A208615.

a(n) ~ c * 256^n / n^(15/2), where c = 1.536590923866647845196812662963243246... . - Vaclav Kotesovec, Sep 03 2014

A208631 Number of Young tableaux with n n-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).

Original entry on oeis.org

1, 1, 3, 53, 25187, 705002611, 1672481205752413, 475092942773985252468181, 22081439406257212482754663652213531, 220381419513554767061883905294847700173775763891, 599868749018773480515945947095662848011697924400242771204050409

Offset: 0

Alois P. Heinz, Feb 29 2012

Also the number of (n^2-1)-step walks on n-dimensional cubic lattice from (1,0,...,0) to (n,n,...,n) with positive unit steps in all dimensions such that for each point (p_1,p_2,...,p_n) we have p_1<=p_2<=...<=p_n or p_1>=p_2>=...>=p_n.

Alois P. Heinz, Table of n, a(n) for n = 0..12

Main diagonal of A208615.

A208729 Number of Young tableaux with i k-length rows with i,k>=0, i+k=n, increasing entries down the columns and monotonic entries along the rows (first row increasing).

Original entry on oeis.org

1, 2, 3, 4, 7, 20, 107, 1251, 39449, 3601484, 993083163, 822645013440, 2233613397459767, 19448649149110190799, 611288282025228989179209, 65375294476542363327381312458, 27613527789685567969428106708416272, 41649724056091694466822995563486395949185

Offset: 0

Alois P. Heinz, Mar 01 2012

a(n) is also the number of (i*k-1)-step walks (for all i,k>=0, i+k=n) on k-dimensional cubic lattice from (1,0,...,0) to (i,i,...,i) with positive unit steps in all dimensions such that for each point (p_1,p_2,...,p_k) we have p_1<=p_2<=...<=p_k or p_1>=p_2>=...>=p_k.

Alois P. Heinz, Table of n, a(n) for n = 0..25

Antidiagonal sums of A208615.

A208617 Number of Young tableaux with 4 n-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).

Original entry on oeis.org

1, 1, 35, 587, 25187, 1676707, 140422657, 13675362559, 1489926719139, 177296325559211, 22661600612752505, 3073259866183533755, 438091469007903238421, 65166105272787401522141, 10056663348255976399237441, 1602608180008201242503733271

Offset: 0

Alois P. Heinz, Feb 29 2012

Also the number of (4*n-1)-step walks on n-dimensional cubic lattice from (1,0,...,0) to (4,4,...,4) with positive unit steps in all dimensions such that for each point (p_1,p_2,...,p_n) we have p_1<=p_2<=...<=p_n or p_1>=p_2>=...>=p_n.

Row n=4 of A208615.

A208618 Number of Young tableaux with 5 n-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).

Original entry on oeis.org

1, 1, 126, 7572, 1725819, 705002611, 396803649991, 278635710716650, 231474950997766763, 219738417947792525211, 232553597317851557785623, 269396684883944249352055973, 336839101974197524267892335361, 449620757900366812848744648452561

Offset: 0

Alois P. Heinz, Feb 29 2012

Also the number of (5*n-1)-step walks on n-dimensional cubic lattice from (1,0,...,0) to (5,5,...,5) with positive unit steps in all dimensions such that for each point (p_1,p_2,...,p_n) we have p_1<=p_2<=...<=p_n or p_1>=p_2>=...>=p_n.

Row n=5 of A208615.

A208619 Number of Young tableaux with 6 n-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).

Original entry on oeis.org

1, 1, 462, 109027, 144558247, 398084427253, 1672481205752413, 9490918987253894191, 67868136936393109678363, 583693245266271046705306483, 5838544884938502473966453328313, 66244125517281822956796820132971163, 836288765056123179126895804194418164733

Offset: 0

Alois P. Heinz, Feb 29 2012

Also the number of (6*n-1)-step walks on n-dimensional cubic lattice from (1,0,...,0) to (6,6,...,6) with positive unit steps in all dimensions such that for each point (p_1,p_2,...,p_n) we have p_1<=p_2<=...<=p_n or p_1>=p_2>=...>=p_n.

Row n=6 of A208615.

A208620 Number of Young tableaux with 7 n-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).

Original entry on oeis.org

1, 1, 1716, 1705249, 14029729645, 279481714446151, 9493821912766657291, 475092942773985252468181, 32103240681864904236146331299, 2760173043757661872972723537937635, 289232902027154515366683463668541370431, 35764586048631587795405572631302247852797701

Offset: 0

Alois P. Heinz, Feb 29 2012

Also the number of (7*n-1)-step walks on n-dimensional cubic lattice from (1,0,...,0) to (7,7,...,7) with positive unit steps in all dimensions such that for each point (p_1,p_2,...,p_n) we have p_1<=p_2<=...<=p_n or p_1>=p_2>=...>=p_n.

Row n=7 of A208615.

A208621 Number of Young tableaux with 8 n-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).

Original entry on oeis.org

1, 1, 6435, 28440320, 1523926182363, 232075055225078521, 67887185669916054862201, 32104063492616280061833179530, 22081439406257212482754663652213531, 20535540740510211632088991774438342144131, 24486820402563168156475227361324722817780058649

Offset: 0

Alois P. Heinz, Feb 29 2012

Also the number of (8*n-1)-step walks on n-dimensional cubic lattice from (1,0,...,0) to (8,8,...,8) with positive unit steps in all dimensions such that for each point (p_1,p_2,...,p_n) we have p_1<=p_2<=...<=p_n or p_1>=p_2>=...>=p_n.

Row n=8 of A208615.

cp's OEIS Frontend

A185148 Number of rectangular arrangements of [1,3n] in 3 increasing sequences of size n and n monotonic sequences of size 3.

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A208616 Number of Young tableaux with 3 n-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).

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A208624 Number of Young tableaux with n 4-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).

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A208631 Number of Young tableaux with n n-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).

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A208729 Number of Young tableaux with i k-length rows with i,k>=0, i+k=n, increasing entries down the columns and monotonic entries along the rows (first row increasing).

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A208617 Number of Young tableaux with 4 n-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).

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A208618 Number of Young tableaux with 5 n-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).

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A208619 Number of Young tableaux with 6 n-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).

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A208620 Number of Young tableaux with 7 n-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).

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A208621 Number of Young tableaux with 8 n-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing).

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