[seqfan] Re: number of different digit patterns of n-digit numbers

franktaw at netscape.net franktaw at netscape.net
Fri Aug 28 05:18:02 CEST 2009

```Yes, it is the sum of the first k Stirling numbers of the second kind.

I didn't use that for the PARI program since PARI (at least the version
I currently have) doesn't have a primitive for Stirling numbers.

-----Original Message-----
From: David Wilson <davidwwilson at comcast.net>

Would this sequence have a formula in terms of Stirling number of the
second
kind?

----- Original Message -----
From: <franktaw at netscape.net>
To: <seqfan at list.seqfan.eu>
Sent: Thursday, August 27, 2009 6:44 PM
Subject: [seqfan] Re: number of different digit patterns of n-digit
numbers

> This is just the number of set partitions of n into at most 10 parts.
> The condition that the first ball not go in the first box is not
> relevant; you can always permute the digits so that the first one is
> not zero.
>
> Sequences of this type are in the OEIS for up to 8 (i.e., set
> partitions into at most 8 parts, which is A099263).
>
> Here's a PARI program to generate sequences of this type:
>
>
a(n,k)=local(ps);ps=exp(sum(i=1,k,x^i/i!)+x*O(x^n));vector(n,i,polcoeff(p

> s,i)*i!)
>
> Here n is the number of terms you want, and k is the maximum number of
> parts.  For k = 10, we get:
>
> 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678569, 4213584,
> 27644267, 190897305, 1382935569, 10479884654, 82861996310,
> 682044632178, 5832378929502, 51720008131148, 474821737584174,
> 4506150050048604, 44145239041717738, 445876518513670356,
> 4637570299337888742, 49618383871367215282, 545551902886241817684,
> 6158380541703927984540, 71311068810038965177080,
> 846359710104516310431744
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: Tanya Khovanova <mathoflove-seqfan at yahoo.com>
>
> Dear Sequence Fans,
>
> I would like to propose a new sequence:
>
> Number of different digit patterns of n-digit numbers.
> or
> Number of ways to put n labeled balls into 10 indistinguishable boxes
> so that
> the first ball can't go into the first box.
> or
> Number of equivalence classes of n-digit numbers with respect to digit
> permutations.
>
> The sequence starts at Bell numbers: 1,2,5,15,52,203,877,4140,21147,
> but are not
> Bell numbers.
>
> I know that OEIS is on vacation, but I need this sequence for my blog
> essay. So
> I am not sure if I should submit.
>
> Besides, I need more terms to distinguish it from Bell numbers.
>
> What should I do, and can you help?
>
> Tanya

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