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%I #34 Jan 28 2019 17:13:35
%S 1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,6,10,1,1,1,1,15,53,35,1,1,1,1,43,
%T 491,587,126,1,1,1,1,133,6091,25187,7572,462,1,1,1,1,430,87781,
%U 1676707,1725819,109027,1716,1,1,1,1,1431,1386529,140422657,705002611,144558247,1705249,6435,1,1
%N Number of Young tableaux A(n,k) with n k-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing); square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A(n,k) is also the number of (n*k-1)-step walks on k-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_k) we have p_1<=p_2<=...<=p_k or p_1>=p_2>=...>=p_k.
%H Alois P. Heinz, <a href="/A208615/b208615.txt">Antidiagonals n = 0..25, flattened</a>
%e A(2,3) = 6:
%e +---+ +---+ +---+ +---+ +---+ +---+
%e |123| |123| |124| |125| |134| |135|
%e |456| |654| |356| |346| |256| |246|
%e +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
%e |x |100| |x |100| |x |100| |x |100| |x |100| |x |100|
%e | x |110| | x |110| | x |110| | x |110| |x |200| |x |200|
%e | x|111| | x|111| |x |210| |x |210| | x |210| | x |210|
%e |x |211| | x|112| | x|211| | x |220| | x|211| | x |220|
%e | x |221| | x |122| | x |221| | x|221| | x |221| | x|221|
%e | x|222| |x |222| | x|222| | x|222| | x|222| | x|222|
%e +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 3, 6, 15, 43, 133, ...
%e 1, 1, 10, 53, 491, 6091, 87781, ...
%e 1, 1, 35, 587, 25187, 1676707, 140422657, ...
%e 1, 1, 126, 7572, 1725819, 705002611, 396803649991, ...
%e 1, 1, 462, 109027, 144558247, 398084427253, 1672481205752413, ...
%p b:= proc() option remember;
%p `if`(nargs<2, 1, `if`(args[1]=args[nargs],
%p `if`(args[1]=0, 1, 2* b(args[1]-1, seq(args[i], i=2..nargs))),
%p `if`(args[1]>0, b(args[1]-1, seq(args[i], i=2..nargs)), 0)
%p +add(`if`(args[j]>args[j-1], b(seq(args[i] -`if`(i=j, 1, 0)
%p , i=1..nargs)), 0), j=2..nargs) ))
%p end:
%p A:= (n, k)-> `if`(n=0 or k=0, 1, b(n-1, n$(k-1))):
%p seq(seq(A(n, d-n), n=0..d), d=0..12);
%t b[args__] := b[args] = If[(nargs = Length[{args}]) < 2, 1, If[First[{args}] == Last[{args}], If[First[{args}] == 0, 1, 2*b[First[{args}]-1, Sequence @@ Rest[{args}]]], If[First[{args}] > 0, b[First[{args}]-1, Sequence @@ Rest[{args}]], 0] + Sum [If[{args}[[j]] > {args}[[j-1]], b[Sequence @@ Table[{args}[[i]] - If[i == j, 1, 0], {i, 1, nargs}]], 0], {j, 2, nargs}] ] ]; a[n_, k_] := If[n == 0 || k == 0, 1, b[n-1, Sequence @@ Array[n&, k-1]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 12 2013, translated from Maple *)
%Y Rows 0+1, 2-10 give: A000012, A141351 (for n>1), A208616, A208617, A208618, A208619, A208620, A208621, A208622, A208623.
%Y Columns 0+1, 2-10 give: A000012, A088218, A185148, A208624, A208625, A208626, A208627, A208628, A208629, A208630.
%Y Main diagonal gives: A208631.
%Y Antidiagonal sums give: A208729.
%K nonn,tabl,walk
%O 0,13
%A _Alois P. Heinz_, Feb 29 2012
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