Pith Number

pith:NFWDCNVX

pith:2014:NFWDCNVXBVUF2GIE5BWL4SAIMZ

not attested not anchored not stored refs pending

There are $(r+1)(r+2)(2r+3)(r^2+3r+5)$ Ways For the Four Teams of a World Cup Group to Each Have $r$ Goals For and $r$ Goals Against [Thanks to the Soccer Analog of Prop. 4.6.19 of Richard Stanley's (Classic!) EC1]

Doron Zeilberger, Shalosh B. Ekhad

arxiv:1407.1919 v1 · 2014-07-08 · math.CO

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{NFWDCNVXBVUF2GIE5BWL4SAIMZ}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Receipt and verification

First computed	2026-05-18T02:48:08.907765Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

696c3136b70d685d1904e86cbe4808666e47d38764132ad69ff76610c009a64e

Aliases

arxiv: 1407.1919 · arxiv_version: 1407.1919v1 · doi: 10.48550/arxiv.1407.1919 · pith_short_12: NFWDCNVXBVUF · pith_short_16: NFWDCNVXBVUF2GIE · pith_short_8: NFWDCNVX

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/NFWDCNVXBVUF2GIE5BWL4SAIMZ \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 696c3136b70d685d1904e86cbe4808666e47d38764132ad69ff76610c009a64e

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "47b8ed6fb9bb19369f12e717ba00c87b278b91167a239d25e14ec1717d63b39d",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2014-07-08T00:57:10Z",
    "title_canon_sha256": "407f65ef1d65aa44efb000bccf4261df7e642347ed4a5bc1545c2181e31a898c"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1407.1919",
    "kind": "arxiv",
    "version": 1
  }
}