A296560 -id:A296560 - cp's OEIS frontend

A296188 Number of normal semistandard Young tableaux whose shape is the integer partition with Heinz number n.

1, 1, 2, 1, 4, 4, 8, 1, 6, 12, 16, 6, 32, 32, 28, 1, 64, 16, 128, 24, 96, 80, 256, 8, 44, 192, 22, 80, 512, 96, 1024, 1, 288, 448, 224, 30, 2048, 1024, 800, 40, 4096, 400, 8192, 240, 168, 2304, 16384, 10, 360, 204, 2112, 672, 32768, 68, 832, 160, 5376, 5120

Offset: 1

Gus Wiseman, Feb 14 2018

nonn

A tableau is normal if its entries span an initial interval of positive integers. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(9) = 6 tableaux:
1 3   1 2   1 2   1 2   1 1   1 1
2 4   3 4   3 3   2 3   2 3   2 2

Richard P. Stanley, Enumerative Combinatorics Volume 2, Cambridge University Press, 1999, Chapter 7.10.

FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux

Cf. A000085, A001222, A056239, A063834, A112798, A122111, A138178, A153452, A191714, A210391, A228125, A296150, A296560, A296561, A299202, A299966, A300056, A300121.

conj[y_List]:=If[Length[y]===0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
conj[n_Integer]:=Times@@Prime/@conj[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]];
ssyt[n_]:=If[n===1,1,Sum[ssyt[n/q*Times@@Cases[FactorInteger[q],{p_,k_}:>If[p===2,1,NextPrime[p,-1]^k]]],{q,Rest[Divisors[n]]}]];
Table[ssyt[conj[n]],{n,50}]

Let b(n) = Sum_{d|n, d>1} b(n * d' / d) where if d = Product_i prime(s_i)^m(i) then d' = Product_i prime(s_i - 1)^m(i) and prime(0) = 1. Then a(n) = b(conj(n)) where conj = A122111.

A317545 Number of multimin factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 5, 1, 2, 2, 8, 1, 4, 1, 5, 2, 2, 1, 12, 2, 2, 4, 5, 1, 5, 1, 16, 2, 2, 2, 11, 1, 2, 2, 12, 1, 5, 1, 5, 5, 2, 1, 28, 2, 4, 2, 5, 1, 8, 2, 12, 2, 2, 1, 15, 1, 2, 5, 32, 2, 5, 1, 5, 2, 5, 1, 29, 1, 2, 4, 5, 2, 5, 1, 28, 8, 2, 1, 15, 2, 2, 2, 12, 1, 12, 2, 5, 2, 2, 2, 64, 1, 4, 5, 11, 1, 5, 1, 12, 5

Offset: 1

Gus Wiseman, Jul 31 2018

nonn

A multimin factorizations of n is an ordered factorization of n into factors greater than 1 such that the sequence of minimal primes dividing each factor is weakly increasing.

			The a(36) = 11 multimin factorizations:
  (36),
  (2*18), (4*9), (6*6), (12*3), (18*2),
  (2*2*9), (2*6*3), (4*3*3), (6*2*3),
  (2*2*3*3).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A007716, A020639, A034691, A255397, A296560, A300335, A317546.

a[n_]:=If[n==1,1,Sum[a[d],{d,Divisors[n/FactorInteger[n][[1,1]]]}]];
Array[a,100]

A317545(n) = if(1==n,1,my(spf = factor(n)[1,1]); sumdiv(n/spf,d,A317545(d))); \\ Antti Karttunen, Sep 10 2018

memo317545 = Map(); \\ Memoized version.
A317545(n) = if(1==n,1,if(mapisdefined(memo317545, n), mapget(memo317545, n), my(spf = factor(n)[1,1], v = sumdiv(n/spf,d,A317545(d))); mapput(memo317545, n, v); (v))); \\ Antti Karttunen, Sep 10 2018

a(1) = 1; a(n > 1) = Sum_{d|(n/p)} a(d), where p is the smallest prime dividing n.

More terms from Antti Karttunen, Sep 10 2018

cp's OEIS Frontend

A296188 Number of normal semistandard Young tableaux whose shape is the integer partition with Heinz number n.

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