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 A264033 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A130519(n+1)) is the number of integer partitions of n having k pairs of different size. 2
 1, 1, 2, 2, 1, 3, 1, 1, 2, 2, 2, 1, 4, 2, 1, 2, 2, 2, 3, 3, 3, 1, 2, 1, 4, 3, 3, 3, 2, 3, 2, 1, 1, 3, 4, 3, 5, 2, 5, 2, 2, 1, 2, 1, 4, 4, 4, 7, 3, 4, 2, 4, 5, 1, 0, 2, 2, 2, 5, 5, 8, 2, 9, 4, 4, 3, 4, 1, 4, 1, 1, 2, 1, 6, 5, 4, 9, 4, 9, 4, 6, 5, 7, 2, 4, 3, 1, 2, 2, 2, 1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Row sums give A000041. T(n,0) gives A000005(n) for n>0. - Alois P. Heinz, Nov 01 2015 REFERENCES Richard Stanley, Enumerative combinatorics. Vol. 2 MathSciNet:1676282, page 375. LINKS Alois P. Heinz, Rows n = 0..80, flattened FindStat - Combinatorial Statistic Finder, Degree of the polynomial counting the number of semistandard Young-tableaux of shape k*lambda. FORMULA Sum_{k>0} k * T(n,k) = A271370(n). - Alois P. Heinz, Apr 05 2016 EXAMPLE Triangle begins: 1; 1; 2; 2,1; 3,1,1; 2,2,2,1; 4,2,1,2,2; 2,3,3,3,1,2,1; 4,3,3,3,2,3,2,1,1; 3,4,3,5,2,5,2,2,1,2,1; 4,4,4,7,3,4,2,4,5,1,0,2,2; 2,5,5,8,2,9,4,4,3,4,1,4,1,1,2,1; 6,5,4,9,4,9,4,6,5,7,2,4,3,1,2,2,2,1,1; ... MAPLE b:= proc(n, i, p, t) option remember; expand(       `if`(n=0, x^t, `if`(i<1, 0, add(        b(n-i*j, i-1, p+j, t+j*p), j=0..n/i))))     end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n\$2, 0\$2)): seq(T(n), n=0..15);  # Alois P. Heinz, Nov 01 2015 MATHEMATICA b[n_, i_, p_, t_] := b[n, i, p, t] = Expand[If[n==0, x^t, If[i<1, 0, Sum[b[n-i*j, i-1, p+j, t+j*p], {j, 0, n/i}]]]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0, 0]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *) CROSSREFS Cf. A000005, A000041, A130519, A271370. Sequence in context: A193262 A120967 A116687 * A236293 A056044 A236097 Adjacent sequences:  A264030 A264031 A264032 * A264034 A264035 A264036 KEYWORD nonn,tabf AUTHOR Christian Stump, Nov 01 2015 STATUS approved

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Last modified October 1 04:06 EDT 2020. Contains 337441 sequences. (Running on oeis4.)